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Theorem fvmpt3i 6749
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 6748 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  Vcvv 3473  cmpt 5122  cfv 6331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339
This theorem is referenced by:  isf32lem9  9761  axcc2lem  9836  caucvg  15015  ismre  16840  mrisval  16880  frmdup1  18008  frmdup2  18009  qusghm  18374  pmtrfval  18557  odf1  18668  vrgpfval  18871  dprdz  19131  dmdprdsplitlem  19138  dprd2dlem2  19141  dprd2dlem1  19142  dprd2da  19143  ablfac1a  19170  ablfac1b  19171  ablfac1eu  19174  ipdir  20759  ipass  20765  isphld  20774  istopon  21496  qustgpopn  22704  qustgplem  22705  tcphcph  23820  cmvth  24573  mvth  24574  dvle  24589  lhop1  24596  dvfsumlem3  24610  pige3ALT  25091  fsumdvdscom  25749  logfacbnd3  25786  dchrptlem1  25827  dchrptlem2  25828  lgsdchrval  25917  dchrisumlem3  26054  dchrisum0flblem1  26071  dchrisum0fno1  26074  dchrisum0lem1b  26078  dchrisum0lem2a  26080  dchrisum0lem2  26081  logsqvma2  26106  log2sumbnd  26107  measdivcst  31491  measdivcstALTV  31492  mrexval  32756  mexval  32757  mdvval  32759  msubvrs  32815  mthmval  32830  f1omptsnlem  34634  upixp  35043  ismrer1  35152  frlmsnic  39269  uzmptshftfval  40833  amgmwlem  45090  amgmlemALT  45091
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