Theorem mpteq12 4011

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 1506	. 2 |- ( A = C -> A. x A = C )
2		mpteq12f 4008	. 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 1329   = wceq 1331   A.wral 2416   |-> cmpt 3989

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-opab 3990  df-mpt 3991

This theorem is referenced by:  mpteq1  4012  mpteqb  5511  fmptcof  5587  mapxpen  6742  prodeq2w  11325