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Theorem csbima12 5518
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 3569 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(𝐹𝐵))
2 csbeq1 3569 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3569 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3imaeq12d 5502 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
51, 4eqeq12d 2666 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
6 vex 3234 . . . 4 𝑦 ∈ V
7 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝐵
97, 8nfima 5509 . . . 4 𝑥(𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
10 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11imaeq12d 5502 . . . 4 (𝑥 = 𝑦 → (𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵))
136, 9, 12csbief 3591 . . 3 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
145, 13vtoclg 3297 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
15 csbprc 4013 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
16 csbprc 4013 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
1716imaeq2d 5501 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐴 / 𝑥𝐹 “ ∅))
18 ima0 5516 . . . 4 (𝐴 / 𝑥𝐹 “ ∅) = ∅
1917, 18syl6req 2702 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
2015, 19eqtrd 2685 . 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 1523  wcel 2030  Vcvv 3231  csb 3566  c0 3948  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  csbrn  5631  disjpreima  29523  csbpredg  33302  brtrclfv2  38336  sbcheg  38390  csbfv12gALTVD  39449
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