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Theorem csbima12 6097
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 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(𝐹𝐵))
2 csbeq1 3902 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3902 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3imaeq12d 6079 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
51, 4eqeq12d 2753 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
6 vex 3484 . . . 4 𝑦 ∈ V
7 nfcsb1v 3923 . . . . 5 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3923 . . . . 5 𝑥𝑦 / 𝑥𝐵
97, 8nfima 6086 . . . 4 𝑥(𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
10 csbeq1a 3913 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3913 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11imaeq12d 6079 . . . 4 (𝑥 = 𝑦 → (𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵))
136, 9, 12csbief 3933 . . 3 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
145, 13vtoclg 3554 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
15 csbprc 4409 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
16 csbprc 4409 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
1716imaeq2d 6078 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐴 / 𝑥𝐹 “ ∅))
18 ima0 6095 . . . 4 (𝐴 / 𝑥𝐹 “ ∅) = ∅
1917, 18eqtr2di 2794 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2015, 19eqtrd 2777 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2114, 20pm2.61i 182 1 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  c0 4333  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  csbrn  6223  csbpredg  6327  disjpreima  32597  brtrclfv2  43740  sbcheg  43792  csbfv12gALTVD  44919
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