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Theorem mpteq2da 4229
Description: Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq2da.1  |-  F/ x ph
mpteq2da.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
mpteq2da  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2381 . . 3  |-  A  =  A
21ax-gen 1552 . 2  |-  A. x  A  =  A
3 mpteq2da.1 . . 3  |-  F/ x ph
4 mpteq2da.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
54ex 424 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
63, 5ralrimi 2724 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
7 mpteq12f 4220 . 2  |-  ( ( A. x  A  =  A  /\  A. x  e.  A  B  =  C )  ->  (
x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
82, 6, 7sylancr 645 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1717   A.wral 2643    e. cmpt 4201
This theorem is referenced by:  mpteq2dva  4230  sumeq1f  12403  sumeq2ii  12408  xkocnv  17761  utopsnneiplem  18192  offval2f  23916  esumf1o  24235  prodeq1f  25007  prodeq2ii  25012  mzpsubmpt  26485  mzpexpmpt  26487  refsum2cnlem1  27370  fmuldfeqlem1  27374  stoweidlem2  27413  stoweidlem6  27417  stoweidlem8  27419  stoweidlem17  27428  stoweidlem19  27430  stoweidlem20  27431  stoweidlem21  27432  stoweidlem22  27433  stoweidlem23  27434  stoweidlem32  27443  stoweidlem36  27447  stoweidlem40  27451  stoweidlem41  27452  stoweidlem47  27458  stirlinglem15  27499
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 2362
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 2368  df-cleq 2374  df-clel 2377  df-ral 2648  df-opab 4202  df-mpt 4203
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