Theorem mpteq12 3840

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 1419	. 2 |- ( A = C -> A. x A = C )
2		mpteq12f 3837	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
3	1, 2	sylan 267	1 |- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   A.wral 2306   |-> cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820

This theorem is referenced by:  mpteq1  3841  mpteqb  5261  fmptcof  5331