Theorem csbima12g 4686

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
csbima12g	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbima12g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2855	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵))
2		csbeq1 2855	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 2855	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	imaeq12d 4669	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2054	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)))
6		vex 2560	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 2882	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 2882	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfima 4676	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 2860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 2860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	imaeq12d 4669	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 2891	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 2613	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ⦋csb 2852   “ cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)