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Theorem mpteq12 4115
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 1606 . 2  |-  ( A  =  C  ->  A. x  A  =  C )
2 mpteq12f 4112 . 2  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
31, 2sylan 457 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 358   A.wal 1530    = wceq 1632   A.wral 2556    e. cmpt 4093
This theorem is referenced by:  mpteq1  4116  mpteqb  5630  fmptcof  5708  mapxpen  7043  sumeq2w  12181  prdsdsval2  13399  prdsdsval3  13400  ablfac2  15340  xkocnv  17521  voliun  18927  itgeq1f  19142  itgeq2  19148  iblcnlem  19159  esumeq2  23433  esumcvg  23469  cndprobprob  23656  cprodeq2w  24134  bddiblnc  25021  stoweidlem17  27869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-opab 4094  df-mpt 4095
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