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Theorem fvmpt3i 5621
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 10 . 2  |-  ( x  e.  D  ->  B  e.  _V )
51, 2, 4fvmpt3 5620 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  isf32lem9  8003  axcc2lem  8078  caucvg  12167  ismre  13508  mrisval  13548  frmdup1  14502  frmdup2  14503  divsghm  14735  odf1  14891  vrgpfval  15091  dprdz  15281  dmdprdsplitlem  15288  dprd2dlem2  15291  dprd2dlem1  15292  dprd2da  15293  ablfac1a  15320  ablfac1b  15321  ablfac1eu  15324  ipdir  16559  ipass  16565  isphld  16574  istopon  16679  divstgpopn  17818  divstgplem  17819  tchcph  18683  cmvth  19354  mvth  19355  dvle  19370  lhop1  19377  dvfsumlem3  19391  pige3  19901  fsumdvdscom  20441  logfacbnd3  20478  dchrptlem1  20519  dchrptlem2  20520  lgsdchrval  20602  dchrisumlem3  20656  dchrisum0flblem1  20673  dchrisum0fno1  20676  dchrisum0lem1b  20680  dchrisum0lem2a  20682  dchrisum0lem2  20683  logsqvma2  20708  log2sumbnd  20709  measdivcstOLD  23566  measdivcst  23567  upixp  26506  ismrer1  26665  pmtrfval  27496
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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