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Theorem fvmpt3i 7004
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 = 𝐴𝐵 = 𝐶)
fvmpt3.b 𝐹 = (𝑥𝐷𝐵)
fvmpt3i.c 𝐵 ∈ V
Assertion
Ref Expression
fvmpt3i (𝐴𝐷 → (𝐹𝐴) = 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 = 𝐴𝐵 = 𝐶)
2 fvmpt3.b . 2 𝐹 = (𝑥𝐷𝐵)
3 fvmpt3i.c . . 3 𝐵 ∈ V
43a1i 11 . 2 (𝑥𝐷𝐵 ∈ V)
51, 2, 4fvmpt3 7003 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cmpt 5232  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552
This theorem is referenced by:  isf32lem9  10356  axcc2lem  10431  caucvg  15625  ismre  17534  mrisval  17574  frmdup1  18745  frmdup2  18746  qusghm  19129  pmtrfval  19318  odf1  19430  vrgpfval  19634  dprdz  19900  dmdprdsplitlem  19907  dprd2dlem2  19910  dprd2dlem1  19911  dprd2da  19912  ablfac1a  19939  ablfac1b  19940  ablfac1eu  19943  ipdir  21192  ipass  21198  isphld  21207  istopon  22414  qustgpopn  23624  qustgplem  23625  tcphcph  24754  cmvth  25508  mvth  25509  dvle  25524  lhop1  25531  dvfsumlem3  25545  pige3ALT  26029  fsumdvdscom  26689  logfacbnd3  26726  dchrptlem1  26767  dchrptlem2  26768  lgsdchrval  26857  dchrisumlem3  26994  dchrisum0flblem1  27011  dchrisum0fno1  27014  dchrisum0lem1b  27018  dchrisum0lem2a  27020  dchrisum0lem2  27021  logsqvma2  27046  log2sumbnd  27047  measdivcst  33253  measdivcstALTV  33254  mrexval  34523  mexval  34524  mdvval  34526  msubvrs  34582  mthmval  34597  gg-cmvth  35212  f1omptsnlem  36265  upixp  36645  ismrer1  36754  frlmsnic  41158  fsuppind  41210  uzmptshftfval  43153  amgmwlem  47897  amgmlemALT  47898
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