Metamath Proof Explorer


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 )