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Mirrors > Home > ILE Home > Th. List > xpssres | GIF version |
Description: Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
xpssres | ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4357 | . . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
2 | inxp 4470 | . . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
3 | incom 3129 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴) | |
4 | inv1 3253 | . . . 4 ⊢ (𝐵 ∩ V) = 𝐵 | |
5 | 3, 4 | xpeq12i 4367 | . . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐶 ∩ 𝐴) × 𝐵) |
6 | 1, 2, 5 | 3eqtri 2064 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶 ∩ 𝐴) × 𝐵) |
7 | df-ss 2931 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶) | |
8 | 7 | biimpi 113 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶) |
9 | 8 | xpeq1d 4368 | . 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐶 ∩ 𝐴) × 𝐵) = (𝐶 × 𝐵)) |
10 | 6, 9 | syl5eq 2084 | 1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 Vcvv 2557 ∩ cin 2916 ⊆ wss 2917 × 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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
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-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-rel 4352 df-res 4357 |
This theorem is referenced by: (None) |
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