Theorem rexrnmpo

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypotheses	rngop.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
	ralrnmpo.2	$$⊢z=C\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rexrnmpo	$$⊢\forall x\in A\forall y\in BC\in V\to \left(\exists z\in \mathrm{ran}F\phi \leftrightarrow \exists x\in A\exists y\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rngop.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
2		ralrnmpo.2	$$⊢z=C\to \left(\phi \leftrightarrow \psi \right)$$
3	2	notbid	$$⊢z=C\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
4	1 3	ralrnmpo	$$⊢\forall x\in A\forall y\in BC\in V\to \left(\forall z\in \mathrm{ran}F\neg \phi \leftrightarrow \forall x\in A\forall y\in B\neg \psi \right)$$
5	4	notbid	$$⊢\forall x\in A\forall y\in BC\in V\to \left(\neg \forall z\in \mathrm{ran}F\neg \phi \leftrightarrow \neg \forall x\in A\forall y\in B\neg \psi \right)$$
6		dfrex2	$$⊢\exists z\in \mathrm{ran}F\phi \leftrightarrow \neg \forall z\in \mathrm{ran}F\neg \phi$$
7		dfrex2	$$⊢\exists y\in B\psi \leftrightarrow \neg \forall y\in B\neg \psi$$
8	7	rexbii	$$⊢\exists x\in A\exists y\in B\psi \leftrightarrow \exists x\in A\neg \forall y\in B\neg \psi$$
9		rexnal	$$⊢\exists x\in A\neg \forall y\in B\neg \psi \leftrightarrow \neg \forall x\in A\forall y\in B\neg \psi$$
10	8 9	bitri	$$⊢\exists x\in A\exists y\in B\psi \leftrightarrow \neg \forall x\in A\forall y\in B\neg \psi$$
11	5 6 10	3bitr4g	$$⊢\forall x\in A\forall y\in BC\in V\to \left(\exists z\in \mathrm{ran}F\phi \leftrightarrow \exists x\in A\exists y\in B\psi \right)$$