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Theorem fvmpt3i 6326
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 6325 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  isf32lem9  9221  axcc2lem  9296  caucvg  14453  ismre  16297  mrisval  16337  frmdup1  17448  frmdup2  17449  qusghm  17744  pmtrfval  17916  odf1  18025  vrgpfval  18225  dprdz  18475  dmdprdsplitlem  18482  dprd2dlem2  18485  dprd2dlem1  18486  dprd2da  18487  ablfac1a  18514  ablfac1b  18515  ablfac1eu  18518  ipdir  20032  ipass  20038  isphld  20047  istopon  20765  qustgpopn  21970  qustgplem  21971  tchcph  23082  cmvth  23799  mvth  23800  dvle  23815  lhop1  23822  dvfsumlem3  23836  pige3  24314  fsumdvdscom  24956  logfacbnd3  24993  dchrptlem1  25034  dchrptlem2  25035  lgsdchrval  25124  dchrisumlem3  25225  dchrisum0flblem1  25242  dchrisum0fno1  25245  dchrisum0lem1b  25249  dchrisum0lem2a  25251  dchrisum0lem2  25252  logsqvma2  25277  log2sumbnd  25278  sgnsv  29855  measdivcstOLD  30415  measdivcst  30416  mrexval  31524  mexval  31525  mdvval  31527  msubvrs  31583  mthmval  31598  f1omptsnlem  33313  upixp  33654  ismrer1  33767  uzmptshftfval  38862  amgmwlem  42876  amgmlemALT  42877
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