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Theorem fvmpt3i 5801
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 5800 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258   ` cfv 5446
This theorem is referenced by:  isf32lem9  8233  axcc2lem  8308  caucvg  12464  ismre  13807  mrisval  13847  frmdup1  14801  frmdup2  14802  divsghm  15034  odf1  15190  vrgpfval  15390  dprdz  15580  dmdprdsplitlem  15587  dprd2dlem2  15590  dprd2dlem1  15591  dprd2da  15592  ablfac1a  15619  ablfac1b  15620  ablfac1eu  15623  ipdir  16862  ipass  16868  isphld  16877  istopon  16982  divstgpopn  18141  divstgplem  18142  tchcph  19186  cmvth  19867  mvth  19868  dvle  19883  lhop1  19890  dvfsumlem3  19904  pige3  20417  fsumdvdscom  20962  logfacbnd3  20999  dchrptlem1  21040  dchrptlem2  21041  lgsdchrval  21123  dchrisumlem3  21177  dchrisum0flblem1  21194  dchrisum0fno1  21197  dchrisum0lem1b  21201  dchrisum0lem2a  21203  dchrisum0lem2  21204  logsqvma2  21229  log2sumbnd  21230  measdivcstOLD  24570  measdivcst  24571  upixp  26422  ismrer1  26538  pmtrfval  27361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454
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