Theorem csbima12 5984

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⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbima12

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3839	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵))
2		csbeq1 3839	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3839	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	imaeq12d 5967	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2755	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)))
6		vex 3434	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3861	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3861	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfima 5974	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3850	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3850	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	imaeq12d 5967	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3871	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 3503	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
15		csbprc 4345	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ∅)
16		csbprc 4345	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
17	16	imaeq2d 5966	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ∅))
18		ima0 5982	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ∅) = ∅
19	17, 18	eqtr2di 2796	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
20	15, 19	eqtrd 2779	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
21	14, 20	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⦋csb 3836  ∅c0 4261   “ cima 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601

This theorem is referenced by:  csbrn  6103  csbpredg  6205  disjpreima  30902  brtrclfv2  41288  sbcheg  41340  csbfv12gALTVD  42472