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Theorem fvmpt3i 5768
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 5767 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226   ` cfv 5413
This theorem is referenced by:  isf32lem9  8197  axcc2lem  8272  caucvg  12427  ismre  13770  mrisval  13810  frmdup1  14764  frmdup2  14765  divsghm  14997  odf1  15153  vrgpfval  15353  dprdz  15543  dmdprdsplitlem  15550  dprd2dlem2  15553  dprd2dlem1  15554  dprd2da  15555  ablfac1a  15582  ablfac1b  15583  ablfac1eu  15586  ipdir  16825  ipass  16831  isphld  16840  istopon  16945  divstgpopn  18102  divstgplem  18103  tchcph  19147  cmvth  19828  mvth  19829  dvle  19844  lhop1  19851  dvfsumlem3  19865  pige3  20378  fsumdvdscom  20923  logfacbnd3  20960  dchrptlem1  21001  dchrptlem2  21002  lgsdchrval  21084  dchrisumlem3  21138  dchrisum0flblem1  21155  dchrisum0fno1  21158  dchrisum0lem1b  21162  dchrisum0lem2a  21164  dchrisum0lem2  21165  logsqvma2  21190  log2sumbnd  21191  measdivcstOLD  24531  measdivcst  24532  upixp  26321  ismrer1  26437  pmtrfval  27261
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421
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