Metamath Proof Explorer


Theorem dmmptg

Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013) (Revised by Mario Carneiro, 14-Sep-2013)

Ref Expression
Assertion dmmptg x A B V dom x A B = A

Proof

Step Hyp Ref Expression
1 eqid x A B = x A B
2 1 dmmpt dom x A B = x A | B V
3 elex B V B V
4 3 ralimi x A B V x A B V
5 rabid2 A = x A | B V x A B V
6 4 5 sylibr x A B V A = x A | B V
7 2 6 eqtr4id x A B V dom x A B = A