Theorem mpteq12 4072

Description: An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)

Assertion

Ref	Expression
mpteq12	|- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   D( x)

Proof of Theorem mpteq12

Step	Hyp	Ref	Expression
1		ax-17 1519	. 2 |- ( A = C -> A. x A = C )
2		mpteq12f 4069	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
3	1, 2	sylan 281	1 |- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   A.wral 2448   |-> cmpt 4050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4051  df-mpt 4052

This theorem is referenced by:  mpteq1  4073  mpteqb  5586  fmptcof  5663  mapxpen  6826  prodeq2w  11519