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Metamath Proof Explorer

Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) Remove dependency on ax-10 . (Revised by SN, 11-Nov-2024)

Ref Expression
Hypotheses mpteq12da.1 
|- F/ x ph
mpteq12da.2 
|- ( ph -> A = C )
mpteq12da.3 
|- ( ( ph /\ x e. A ) -> B = D )
Assertion mpteq12da 
|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof

Step Hyp Ref Expression
1 mpteq12da.1 
 |-  F/ x ph
2 mpteq12da.2 
 |-  ( ph -> A = C )
3 mpteq12da.3 
 |-  ( ( ph /\ x e. A ) -> B = D )
4 nfv 
 |-  F/ y ph
5 3 eqeq2d 
 |-  ( ( ph /\ x e. A ) -> ( y = B <-> y = D ) )
6 5 pm5.32da 
 |-  ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
7 2 eleq2d 
 |-  ( ph -> ( x e. A <-> x e. C ) )
8 7 anbi1d 
 |-  ( ph -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
9 6 8 bitrd 
 |-  ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
10 1 4 9 opabbid 
 |-  ( ph -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
11 df-mpt 
 |-  ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
12 df-mpt 
 |-  ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
13 10 11 12 3eqtr4g 
 |-  ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )