Theorem dmmptd

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	dmmptd.a	|- A = ( x e. B |-> C )
	dmmptd.c	|- ( ( ph /\ x e. B ) -> C e. V )
Assertion	dmmptd	|- ( ph -> dom A = B )

Proof

Step	Hyp	Ref	Expression
1		dmmptd.a	|- A = ( x e. B |-> C )
2		dmmptd.c	|- ( ( ph /\ x e. B ) -> C e. V )
3	1	dmmpt	|- dom A = { x e. B | C e. _V }
4	2	elexd	|- ( ( ph /\ x e. B ) -> C e. _V )
5	4	ralrimiva	|- ( ph -> A. x e. B C e. _V )
6		rabid2	|- ( B = { x e. B | C e. _V } <-> A. x e. B C e. _V )
7	5 6	sylibr	|- ( ph -> B = { x e. B | C e. _V } )
8	3 7	eqtr4id	|- ( ph -> dom A = B )