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Theorem fvmpt3i 6449
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 6448 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by:  isf32lem9  9375  axcc2lem  9450  caucvg  14608  ismre  16452  mrisval  16492  frmdup1  17602  frmdup2  17603  qusghm  17898  pmtrfval  18070  odf1  18179  vrgpfval  18379  dprdz  18629  dmdprdsplitlem  18636  dprd2dlem2  18639  dprd2dlem1  18640  dprd2da  18641  ablfac1a  18668  ablfac1b  18669  ablfac1eu  18672  ipdir  20186  ipass  20192  isphld  20201  istopon  20919  qustgpopn  22124  qustgplem  22125  tchcph  23236  cmvth  23953  mvth  23954  dvle  23969  lhop1  23976  dvfsumlem3  23990  pige3  24468  fsumdvdscom  25110  logfacbnd3  25147  dchrptlem1  25188  dchrptlem2  25189  lgsdchrval  25278  dchrisumlem3  25379  dchrisum0flblem1  25396  dchrisum0fno1  25399  dchrisum0lem1b  25403  dchrisum0lem2a  25405  dchrisum0lem2  25406  logsqvma2  25431  log2sumbnd  25432  sgnsv  30036  measdivcstOLD  30596  measdivcst  30597  mrexval  31705  mexval  31706  mdvval  31708  msubvrs  31764  mthmval  31779  f1omptsnlem  33494  upixp  33837  ismrer1  33950  uzmptshftfval  39047  amgmwlem  43061  amgmlemALT  43062
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