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Theorem csbima12 5641
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbima12 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)

Proof of Theorem csbima12
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3677 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(𝐹𝐵))
2 csbeq1 3677 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3677 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3imaeq12d 5625 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
51, 4eqeq12d 2775 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
6 vex 3343 . . . 4 𝑦 ∈ V
7 nfcsb1v 3690 . . . . 5 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3690 . . . . 5 𝑥𝑦 / 𝑥𝐵
97, 8nfima 5632 . . . 4 𝑥(𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
10 csbeq1a 3683 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3683 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11imaeq12d 5625 . . . 4 (𝑥 = 𝑦 → (𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵))
136, 9, 12csbief 3699 . . 3 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
145, 13vtoclg 3406 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
15 csbprc 4123 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
16 csbprc 4123 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
1716imaeq2d 5624 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐴 / 𝑥𝐹 “ ∅))
18 ima0 5639 . . . 4 (𝐴 / 𝑥𝐹 “ ∅) = ∅
1917, 18syl6req 2811 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2015, 19eqtrd 2794 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2114, 20pm2.61i 176 1 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  c0 4058  cima 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  csbrn  5754  disjpreima  29704  csbpredg  33483  brtrclfv2  38521  sbcheg  38575  csbfv12gALTVD  39634
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