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Theorem fvmpts 6944
Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fvmpts.1 𝐹 = (𝑥𝐶𝐵)
Assertion
Ref Expression
fvmpts ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpts
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3852 . 2 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
2 fvmpts.1 . . 3 𝐹 = (𝑥𝐶𝐵)
3 nfcv 2898 . . . 4 𝑦𝐵
4 nfcsb1v 3873 . . . 4 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3863 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5200 . . 3 (𝑥𝐶𝐵) = (𝑦𝐶𝑦 / 𝑥𝐵)
72, 6eqtri 2759 . 2 𝐹 = (𝑦𝐶𝑦 / 𝑥𝐵)
81, 7fvmptg 6939 1 ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2113  csb 3849  cmpt 5179  cfv 6492
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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500
This theorem is referenced by:  fvmptdf  6947  fvmpocurryd  8213  mptnn0fsupp  13920  mptnn0fsuppr  13922  zsum  15641  prodss  15870  fprodser  15872  fprodn0  15902  fprodefsum  16018  pcmpt  16820  issubc  17759  gsummptnn0fz  19915  mptscmfsupp0  20878  gsummoncoe1  22252  fvmptnn04if  22793  prdsdsf  24311  itgparts  26010  dchrisumlema  27455  abfmpeld  32732  abfmpel  32733  cdlemk40  41177  deg1gprod  42394  aomclem6  43301  ellimcabssub0  45863  constlimc  45870  vonn0ioo2  46934  vonn0icc2  46936  dftermo4  49747
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