Theorem csbima12g 4629

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 2849	. . 3 ⊢ (y = A → ⦋y / x⦌(𝐹 “ B) = ⦋A / x⦌(𝐹 “ B))
2		csbeq1 2849	. . . 4 ⊢ (y = A → ⦋y / x⦌𝐹 = ⦋A / x⦌𝐹)
3		csbeq1 2849	. . . 4 ⊢ (y = A → ⦋y / x⦌B = ⦋A / x⦌B)
4	2, 3	imaeq12d 4612	. . 3 ⊢ (y = A → (⦋y / x⦌𝐹 “ ⦋y / x⦌B) = (⦋A / x⦌𝐹 “ ⦋A / x⦌B))
5	1, 4	eqeq12d 2051	. 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 2554	. . 3 ⊢ y ∈ V
7		nfcsb1v 2876	. . . 4 ⊢ Ⅎx⦋y / x⦌𝐹
8		nfcsb1v 2876	. . . 4 ⊢ Ⅎx⦋y / x⦌B
9	7, 8	nfima 4619	. . 3 ⊢ Ⅎx(⦋y / x⦌𝐹 “ ⦋y / x⦌B)
10		csbeq1a 2854	. . . 4 ⊢ (x = y → 𝐹 = ⦋y / x⦌𝐹)
11		csbeq1a 2854	. . . 4 ⊢ (x = y → B = ⦋y / x⦌B)
12	10, 11	imaeq12d 4612	. . 3 ⊢ (x = y → (𝐹 “ B) = (⦋y / x⦌𝐹 “ ⦋y / x⦌B))
13	6, 9, 12	csbief 2885	. 2 ⊢ ⦋y / x⦌(𝐹 “ B) = (⦋y / x⦌𝐹 “ ⦋y / x⦌B)
14	5, 13	vtoclg 2607	1 ⊢ (A ∈ 𝐶 → ⦋A / x⦌(𝐹 “ B) = (⦋A / x⦌𝐹 “ ⦋A / x⦌B))

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ⦋csb 2846   “ cima 4291

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)