Theorem csbfv 6944

Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)

Assertion

Ref	Expression
csbfv	⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)

Distinct variable group:   𝑥,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem csbfv

Step	Hyp	Ref	Expression
1		csbfv2g 6943	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥))
2		csbvarg 4432	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
3	2	fveq2d 6898	. . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴))
4	1, 3	eqtrd 2765	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴))
5		csbprc 4407	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅)
6		fvprc 6886	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
7	5, 6	eqtr4d 2768	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴))
8	4, 7	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3463  ⦋csb 3890  ∅c0 4323  ‘cfv 6547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-dm 5687  df-iota 6499  df-fv 6555

This theorem is referenced by:  mptcoe1fsupp  22144  mptcoe1matfsupp  22735  mp2pm2mplem4  22742  chfacfscmulfsupp  22792  chfacfpmmulfsupp  22796  cpmidpmatlem3  22805  cayhamlem4  22821  cayleyhamilton1  22825  logbmpt  26751  nbgrcl  29205  nbgrnvtx0  29209  iuninc  32409  disjxpin  32436  finixpnum  37165  cdlemkid3N  40492  cdlemkid4  40493  cdlemk39s  40498  mccllem  45065  clnbgrcl  47240  clnbgrnvtx0  47245