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Mirrors > Home > ILE Home > Th. List > csbresg | GIF version |
Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbresg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbing 3144 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))) | |
2 | csbxpg 4421 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)) | |
3 | csbconstg 2864 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌V = V) | |
4 | 3 | xpeq2d 4369 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) |
5 | 2, 4 | eqtrd 2072 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) |
6 | 5 | ineq2d 3138 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))) |
7 | 1, 6 | eqtrd 2072 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))) |
8 | df-res 4357 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
9 | 8 | csbeq2i 2876 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) |
10 | df-res 4357 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) | |
11 | 7, 9, 10 | 3eqtr4g 2097 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⦋csb 2852 ∩ cin 2916 × cxp 4343 ↾ cres 4347 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-in 2924 df-opab 3819 df-xp 4351 df-res 4357 |
This theorem is referenced by: (None) |
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