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Theorem mpteq1 4253
Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem mpteq1
StepHypRef Expression
1 eqidd 2409 . . 3  |-  ( x  e.  A  ->  C  =  C )
21rgen 2735 . 2  |-  A. x  e.  A  C  =  C
3 mpteq12 4252 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  C )  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
42, 3mpan2 653 1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2670    e. cmpt 4230
This theorem is referenced by:  mpteq1d  4254  fmptap  5886  mpt2mpt  6128  mpt2mptsx  6377  mpt2mpts  6378  tposf12  6467  oarec  6768  pwfseq  8499  wunex2  8573  wuncval2  8582  vrmdfval  14760  sylow1  15196  sylow2b  15216  sylow3lem5  15224  sylow3  15226  gsumconst  15491  gsum2d  15505  gsumcom2  15508  mvrfval  16443  mplcoe1  16487  mplcoe2  16489  ply1coe  16643  gsumfsum  16725  restco  17186  cnmpt1t  17654  cnmpt2t  17662  fmval  17932  symgtgp  18088  prdstgpd  18111  evlsval  19897  dfarea  20756  indv  24367  gsumesum  24408  esumlub  24409  sdclem2  26340  pmtrfval  27265  stoweidlem9  27629  swrdltnd  28004
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
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 2395  df-cleq 2401  df-clel 2404  df-ral 2675  df-opab 4231  df-mpt 4232
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