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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 4621 | . . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
2 | inxp 4743 | . . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
3 | incom 3319 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴) | |
4 | inv1 3450 | . . . 4 ⊢ (𝐵 ∩ V) = 𝐵 | |
5 | 3, 4 | xpeq12i 4631 | . . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐶 ∩ 𝐴) × 𝐵) |
6 | 1, 2, 5 | 3eqtri 2195 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶 ∩ 𝐴) × 𝐵) |
7 | df-ss 3134 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶) | |
8 | 7 | biimpi 119 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶) |
9 | 8 | xpeq1d 4632 | . 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐶 ∩ 𝐴) × 𝐵) = (𝐶 × 𝐵)) |
10 | 6, 9 | eqtrid 2215 | 1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 Vcvv 2730 ∩ cin 3120 ⊆ wss 3121 × cxp 4607 ↾ cres 4611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-opab 4049 df-xp 4615 df-rel 4616 df-res 4621 |
This theorem is referenced by: cnconst2 12986 |
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