Theorem dmmptdff

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 21-Dec-2024)

	Ref	Expression
Hypotheses	dmmptdff.x	$$⊢Ⅎx\phi$$
	dmmptdff.1	$$⊢\underset{\_}{Ⅎ}xB$$
	dmmptdff.a	$$⊢A=\left(x\in B⟼C\right)$$
	dmmptdff.c	$$⊢\left(\phi \wedge x\in B\right)\to C\in V$$
Assertion	dmmptdff	$$⊢\phi \to \mathrm{dom}A=B$$

Proof

Step	Hyp	Ref	Expression
1		dmmptdff.x	$$⊢Ⅎx\phi$$
2		dmmptdff.1	$$⊢\underset{\_}{Ⅎ}xB$$
3		dmmptdff.a	$$⊢A=\left(x\in B⟼C\right)$$
4		dmmptdff.c	$$⊢\left(\phi \wedge x\in B\right)\to C\in V$$
5	3	dmmpt	$$⊢\mathrm{dom}A=\left\{x\in B|C\in \mathrm{V}\right\}$$
6	4	elexd	$$⊢\left(\phi \wedge x\in B\right)\to C\in \mathrm{V}$$
7	1 6	ralrimia	$$⊢\phi \to \forall x\in BC\in \mathrm{V}$$
8	2	rabid2f	$$⊢B=\left\{x\in B|C\in \mathrm{V}\right\}\leftrightarrow \forall x\in BC\in \mathrm{V}$$
9	7 8	sylibr	$$⊢\phi \to B=\left\{x\in B|C\in \mathrm{V}\right\}$$
10	5 9	eqtr4id	$$⊢\phi \to \mathrm{dom}A=B$$