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Theorem csbfv 6143
 Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem csbfv
StepHypRef Expression
1 csbfv2g 6142 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 3955 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 6107 . . 3 (𝐴 ∈ V → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2644 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
5 csbprc 3932 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = ∅)
6 fvprc 6097 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
75, 6eqtr4d 2647 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
84, 7pm2.61i 175 1 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ∅c0 3874  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812 This theorem is referenced by:  mptcoe1fsupp  19406  mptcoe1matfsupp  20426  mp2pm2mplem4  20433  chfacfscmulfsupp  20483  chfacfpmmulfsupp  20487  cpmidpmatlem3  20496  cayhamlem4  20512  cayleyhamilton1  20516  logbmpt  24326  iuninc  28761  disjxpin  28783  finixpnum  32564  cdlemkid3N  35239  cdlemkid4  35240  cdlemk39s  35245  mccllem  38664  nbgrcl  40559  nbgrnvtx0  40563
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