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Mirrors > Home > MPE Home > Th. List > csbima12 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
csbima12 | ⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3731 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵)) | |
2 | csbeq1 3731 | . . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹) | |
3 | csbeq1 3731 | . . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 2, 3 | imaeq12d 5684 | . . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
5 | 1, 4 | eqeq12d 2814 | . . 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 3388 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3744 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹 | |
8 | nfcsb1v 3744 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | 7, 8 | nfima 5691 | . . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) |
10 | csbeq1a 3737 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹) | |
11 | csbeq1a 3737 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
12 | 10, 11 | imaeq12d 5684 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)) |
13 | 6, 9, 12 | csbief 3753 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) |
14 | 5, 13 | vtoclg 3453 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
15 | csbprc 4176 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ∅) | |
16 | csbprc 4176 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
17 | 16 | imaeq2d 5683 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ∅)) |
18 | ima0 5698 | . . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ∅) = ∅ | |
19 | 17, 18 | syl6req 2850 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
20 | 15, 19 | eqtrd 2833 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
21 | 14, 20 | pm2.61i 177 | 1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ⦋csb 3728 ∅c0 4115 “ cima 5315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 |
This theorem is referenced by: csbrn 5812 disjpreima 29914 csbpredg 33671 brtrclfv2 38802 sbcheg 38855 csbfv12gALTVD 39895 |
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