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Theorem fvmpts 6536
 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 3760 . 2 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
2 fvmpts.1 . . 3 𝐹 = (𝑥𝐶𝐵)
3 nfcv 2969 . . . 4 𝑦𝐵
4 nfcsb1v 3773 . . . 4 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3766 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4974 . . 3 (𝑥𝐶𝐵) = (𝑦𝐶𝑦 / 𝑥𝐵)
72, 6eqtri 2849 . 2 𝐹 = (𝑦𝐶𝑦 / 𝑥𝐵)
81, 7fvmptg 6531 1 ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ⦋csb 3757   ↦ cmpt 4954  ‘cfv 6127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135 This theorem is referenced by:  fvmptd  6539  fvmpt2curryd  7667  mptnn0fsupp  13098  mptnn0fsuppr  13100  zsum  14833  prodss  15057  fprodser  15059  fprodn0  15089  fprodefsum  15204  pcmpt  15974  issubc  16854  gsummptnn0fz  18742  gsummptnn0fzOLD  18743  mptscmfsupp0  19291  gsummoncoe1  20041  fvmptnn04if  21031  prdsdsf  22549  itgparts  24216  dchrisumlema  25597  abfmpeld  29999  abfmpel  30000  cnfinltrel  33785  cdlemk40  36991  aomclem6  38471  ellimcabssub0  40642  constlimc  40649  vonn0ioo2  41696  vonn0icc2  41698
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