Theorem mpteq12 4117

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 1540	. 2 |- ( A = C -> A. x A = C )
2		mpteq12f 4114	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
3	1, 2	sylan 283	1 |- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   A.wral 2475   |-> cmpt 4095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-opab 4096  df-mpt 4097

This theorem is referenced by:  mpteq1  4118  mpteqb  5655  fmptcof  5732  mapxpen  6918  prodeq2w  11738