Theorem dmmptdf

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	dmmptdf.x	$$⊢Ⅎx\phi$$
	dmmptdf.a	$$⊢A=\left(x\in B⟼C\right)$$
	dmmptdf.c	$$⊢\left(\phi \wedge x\in B\right)\to C\in V$$
Assertion	dmmptdf	$$⊢\phi \to \mathrm{dom}A=B$$

Proof

Step	Hyp	Ref	Expression
1		dmmptdf.x	$$⊢Ⅎx\phi$$
2		dmmptdf.a	$$⊢A=\left(x\in B⟼C\right)$$
3		dmmptdf.c	$$⊢\left(\phi \wedge x\in B\right)\to C\in V$$
4	2	dmmpt	$$⊢\mathrm{dom}A=\left\{x\in B|C\in \mathrm{V}\right\}$$
5	3	elexd	$$⊢\left(\phi \wedge x\in B\right)\to C\in \mathrm{V}$$
6	1 5	ralrimia	$$⊢\phi \to \forall x\in BC\in \mathrm{V}$$
7		rabid2	$$⊢B=\left\{x\in B|C\in \mathrm{V}\right\}\leftrightarrow \forall x\in BC\in \mathrm{V}$$
8	6 7	sylibr	$$⊢\phi \to B=\left\{x\in B|C\in \mathrm{V}\right\}$$
9	4 8	eqtr4id	$$⊢\phi \to \mathrm{dom}A=B$$