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Theorem fvmpt3i 5937
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvmpt3.a  |-  ( x  =  A  ->  B  =  C )
fvmpt3.b  |-  F  =  ( x  e.  D  |->  B )
fvmpt3i.c  |-  B  e. 
_V
Assertion
Ref Expression
fvmpt3i  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpt3i
StepHypRef Expression
1 fvmpt3.a . 2  |-  ( x  =  A  ->  B  =  C )
2 fvmpt3.b . 2  |-  F  =  ( x  e.  D  |->  B )
3 fvmpt3i.c . . 3  |-  B  e. 
_V
43a1i 11 . 2  |-  ( x  e.  D  ->  B  e.  _V )
51, 2, 4fvmpt3 5936 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059    |-> cmpt 4453   ` cfv 5569
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-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577
This theorem is referenced by:  isf32lem9  8773  axcc2lem  8848  caucvg  13650  ismre  15204  mrisval  15244  frmdup1  16356  frmdup2  16357  qusghm  16627  pmtrfval  16799  odf1  16908  vrgpfval  17108  dprdz  17397  dmdprdsplitlem  17404  dmdprdsplitlemOLD  17405  dprd2dlem2  17409  dprd2dlem1  17410  dprd2da  17411  ablfac1a  17440  ablfac1b  17441  ablfac1eu  17444  ipdir  18972  ipass  18978  isphld  18987  istopon  19718  qustgpopn  20910  qustgplem  20911  tchcph  21972  cmvth  22684  mvth  22685  dvle  22700  lhop1  22707  dvfsumlem3  22721  pige3  23202  fsumdvdscom  23842  logfacbnd3  23879  dchrptlem1  23920  dchrptlem2  23921  lgsdchrval  24003  dchrisumlem3  24057  dchrisum0flblem1  24074  dchrisum0fno1  24077  dchrisum0lem1b  24081  dchrisum0lem2a  24083  dchrisum0lem2  24084  logsqvma2  24109  log2sumbnd  24110  sgnsv  28169  measdivcstOLD  28672  measdivcst  28673  mrexval  29713  mexval  29714  mdvval  29716  msubvrs  29772  mthmval  29787  f1omptsnlem  31252  upixp  31502  ismrer1  31616  uzmptshftfval  36099
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