Theorem mpteq12 4473

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-5 1725	. 2 |- ( A = C -> A. x A = C )
2		mpteq12f 4470	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
3	1, 2	sylan 469	1 |- ( ( A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 367   A.wal 1403   = wceq 1405   A.wral 2753   |-> cmpt 4452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-ral 2758  df-opab 4453  df-mpt 4454

This theorem is referenced by:  mpteq1  4474  mpteqb  5947  fmptcof  6043  mapxpen  7720  prodeq2w  13869  prdsdsval2  15096  prdsdsval3  15097  ablfac2  17458  mdetunilem9  19412  mdetmul  19415  xkocnv  20605  voliun  22254  itgeq1f  22468  itgeq2  22474  iblcnlem  22485  esumeq2  28469  esumcvg  28519  dvtan  31418  bddiblnc  31438