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Theorem fvmpt3i 5812
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 5811 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4269   ` cfv 5457
This theorem is referenced by:  isf32lem9  8246  axcc2lem  8321  caucvg  12477  ismre  13820  mrisval  13860  frmdup1  14814  frmdup2  14815  divsghm  15047  odf1  15203  vrgpfval  15403  dprdz  15593  dmdprdsplitlem  15600  dprd2dlem2  15603  dprd2dlem1  15604  dprd2da  15605  ablfac1a  15632  ablfac1b  15633  ablfac1eu  15636  ipdir  16875  ipass  16881  isphld  16890  istopon  16995  divstgpopn  18154  divstgplem  18155  tchcph  19199  cmvth  19880  mvth  19881  dvle  19896  lhop1  19903  dvfsumlem3  19917  pige3  20430  fsumdvdscom  20975  logfacbnd3  21012  dchrptlem1  21053  dchrptlem2  21054  lgsdchrval  21136  dchrisumlem3  21190  dchrisum0flblem1  21207  dchrisum0fno1  21210  dchrisum0lem1b  21214  dchrisum0lem2a  21216  dchrisum0lem2  21217  logsqvma2  21242  log2sumbnd  21243  measdivcstOLD  24583  measdivcst  24584  upixp  26445  ismrer1  26561  pmtrfval  27384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465
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