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| Mirrors > Home > MPE Home > Th. List > csbres | Structured version Visualization version GIF version | ||
| Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.) | 
| Ref | Expression | 
|---|---|
| csbres | ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-res 5696 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
| 2 | 1 | csbeq2i 3906 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) | 
| 3 | csbxp 5784 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) | |
| 4 | csbconstg 3917 | . . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V) | |
| 5 | 4 | xpeq2d 5714 | . . . . . 6 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) | 
| 6 | 3, 5 | eqtrid 2788 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) | 
| 7 | 0xp 5783 | . . . . . . 7 ⊢ (∅ × V) = ∅ | |
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (∅ × V) = ∅) | 
| 9 | csbprc 4408 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅) | |
| 10 | 9 | xpeq1d 5713 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × V) = (∅ × V)) | 
| 11 | csbprc 4408 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = ∅) | |
| 12 | 8, 10, 11 | 3eqtr4rd 2787 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) | 
| 13 | 6, 12 | pm2.61i 182 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V) | 
| 14 | 13 | ineq2i 4216 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) | 
| 15 | csbin 4441 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) | |
| 16 | df-res 5696 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) | |
| 17 | 14, 15, 16 | 3eqtr4i 2774 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) | 
| 18 | 2, 17 | eqtri 2764 | 1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⦋csb 3898 ∩ cin 3949 ∅c0 4332 × cxp 5682 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-res 5696 | 
| This theorem is referenced by: csbfrecsg 8310 csbima12gALTVD 44922 | 
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