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Theorem csbima12 6072
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 3891 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(𝐹𝐵))
2 csbeq1 3891 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3891 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3imaeq12d 6054 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
51, 4eqeq12d 2742 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
6 vex 3472 . . . 4 𝑦 ∈ V
7 nfcsb1v 3913 . . . . 5 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3913 . . . . 5 𝑥𝑦 / 𝑥𝐵
97, 8nfima 6061 . . . 4 𝑥(𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
10 csbeq1a 3902 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3902 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11imaeq12d 6054 . . . 4 (𝑥 = 𝑦 → (𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵))
136, 9, 12csbief 3923 . . 3 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
145, 13vtoclg 3537 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
15 csbprc 4401 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
16 csbprc 4401 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
1716imaeq2d 6053 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐴 / 𝑥𝐹 “ ∅))
18 ima0 6070 . . . 4 (𝐴 / 𝑥𝐹 “ ∅) = ∅
1917, 18eqtr2di 2783 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2015, 19eqtrd 2766 . 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 1533  wcel 2098  Vcvv 3468  csb 3888  c0 4317  cima 5672
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  csbrn  6196  csbpredg  6300  disjpreima  32324  brtrclfv2  43051  sbcheg  43103  csbfv12gALTVD  44233
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