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Mirrors > Home > ILE Home > Th. List > csbima12g | GIF version |
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.) |
Ref | Expression |
---|---|
csbima12g | ⊢ (A ∈ 𝐶 → ⦋A / x⦌(𝐹 “ B) = (⦋A / x⦌𝐹 “ ⦋A / x⦌B)) |
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)) |
Colors of variables: wff set class |
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-bndl 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) |
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