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Theorem fvmpt3i 5945
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 5944 1  |-  ( A  e.  D  ->  ( F `  A )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    |-> cmpt 4498   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587
This theorem is referenced by:  isf32lem9  8730  axcc2lem  8805  caucvg  13450  ismre  14834  mrisval  14874  frmdup1  15848  frmdup2  15849  divsghm  16091  pmtrfval  16264  odf1  16373  vrgpfval  16573  dprdz  16860  dmdprdsplitlem  16867  dmdprdsplitlemOLD  16868  dprd2dlem2  16872  dprd2dlem1  16873  dprd2da  16874  ablfac1a  16903  ablfac1b  16904  ablfac1eu  16907  ipdir  18434  ipass  18440  isphld  18449  istopon  19186  divstgpopn  20346  divstgplem  20347  tchcph  21408  cmvth  22120  mvth  22121  dvle  22136  lhop1  22143  dvfsumlem3  22157  pige3  22636  fsumdvdscom  23182  logfacbnd3  23219  dchrptlem1  23260  dchrptlem2  23261  lgsdchrval  23343  dchrisumlem3  23397  dchrisum0flblem1  23414  dchrisum0fno1  23417  dchrisum0lem1b  23421  dchrisum0lem2a  23423  dchrisum0lem2  23424  logsqvma2  23449  log2sumbnd  23450  measdivcstOLD  27685  measdivcst  27686  upixp  29674  ismrer1  29788
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