Theorem csbima12g 4947

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 3034	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵))
2		csbeq1 3034	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3034	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	imaeq12d 4929	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2172	. 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 2715	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3064	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3064	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfima 4936	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3040	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3040	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	imaeq12d 4929	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3075	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 2772	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  ⦋csb 3031   “ cima 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599

This theorem is referenced by: (None)