2重逆像

$ {f^\gets}^\gets:2^{2^X}\ni\mathcal X\mapsto\Set{B\in2^Y|\exist A\in\mathcal X\forall x\in X:(x\in A\iff f(x)\in B)}\in2^{2^Y}

$ {f^\gets}^\to:2^{2^Y}\ni\mathcal Y\mapsto\Set{A|\exist B\in\mathcal Y\forall x\in X:(x\in A\iff f(x)\in B)}\in2^{2^X}

$ {f^\to}^\gets:2^{2^Y}\ni\mathcal Y\mapsto\Set{A\in2^X|\exist B\in\mathcal Y\forall y:(y\in B\iff\exist x\in A:y=f(x))}\in2^{2^X}

$ {f^\to}^\to:2^{2^X}\ni\mathcal X\mapsto\Set{B|\exist A\in\mathcal X\forall y:(y\in B\iff \exist x\in A:y=f(x))}\in2^{2^Y}

導出

$ {f^\gets}^\gets(\mathcal X)=\Set{B\in2^Y|f^\gets(B)\in\mathcal X}

$ =\Set{B\in2^Y|\exist A\in\mathcal X:A=f^\gets(B)}

$ =\Set{B\in2^Y|\exist A\in\mathcal X\forall x:(x\in A\iff x\in X\land f(x)\in B)}

$ =\Set{B\in2^Y|\exist A\in\mathcal X\forall x\in X:(x\in A\iff f(x)\in B)}

$ {f^\gets}^\to(\mathcal Y)=\Set{A|\exist B\in\mathcal Y:A=f^\gets(B)}

$ =\Set{A|\exist B\in\mathcal Y\forall x:(x\in A\iff x\in X\land f(x)\in B)}

$ =\Set{A|\exist B\in\mathcal Y\forall x\in X:(x\in A\iff f(x)\in B)}

$ {f^\to}^\gets(\mathcal Y)=\Set{A\in2^X| f^\to(A)\in\mathcal Y}

$ =\Set{A\in2^X|\exist B\in\mathcal Y:B=f^\to(A)}

$ =\Set{A\in2^X|\exist B\in\mathcal Y\forall y:(y\in B\iff\exist x\in A:y=f(x))}

$ {f^\to}^\to(\mathcal X)=\Set{B|\exist A\in\mathcal X:B=f^\to(A)}

$ =\Set{B|\exist A\in\mathcal X\forall y:(y\in B\iff\exist x\in A:y=f(x))}

$ B\in{f^\gets}^\gets(\mathcal X)\iff\exist A\in\mathcal X\forall x:(x\in A\iff f(x)\in B)

$ A\in{f^\gets}^\to(\mathcal Y)\iff\exist B\in\mathcal Y\forall x:(x\in A\iff f(x)\in B)

$ \therefore(\exist B\in\mathcal Y:B\in{f^\gets}^\gets(\mathcal X))\iff(\exists A\in\mathcal X:A\in{f^\gets}^\to(\mathcal Y))

$ \therefore{f^\gets}^\gets(\mathcal X)\cap\mathcal Y\neq\varnothing\iff{f^\gets}^\to(\mathcal Y)\cap\mathcal X\neq\varnothing

1重でも何か法則ないかな

$ b\in f^\to(A)\iff\exist a\in A:b=f(a)

$ a\in f^\gets(B)\iff f(a)\in B

$ \therefore(\exist b\in B:b\in f^\to(A))\iff(\exist a\in A:a\in f^\gets(B))

$ \therefore f^\to(A)\cap B\neq\varnothing\iff f^\gets(B)\cap A\neq\varnothing

やっぱりそうだったみたい