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Theorem mpteq12 4231
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
StepHypRef Expression
1 ax-17 1623 . 2  |-  ( A  =  C  ->  A. x  A  =  C )
2 mpteq12f 4228 . 2  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
31, 2sylan 458 1  |-  ( ( A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649   A.wral 2651    e. cmpt 4209
This theorem is referenced by:  mpteq1  4232  mpteqb  5760  fmptcof  5843  mapxpen  7211  sumeq2w  12415  prdsdsval2  13635  prdsdsval3  13636  ablfac2  15576  xkocnv  17769  voliun  19317  itgeq1f  19532  itgeq2  19538  iblcnlem  19549  esumeq2  24231  esumcvg  24274  prodeq2w  25019  bddiblnc  25977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-ral 2656  df-opab 4210  df-mpt 4211
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