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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 4300 | . . 3 ⊢ ((A × B) ↾ 𝐶) = ((A × B) ∩ (𝐶 × V)) | |
2 | inxp 4413 | . . 3 ⊢ ((A × B) ∩ (𝐶 × V)) = ((A ∩ 𝐶) × (B ∩ V)) | |
3 | incom 3123 | . . . 4 ⊢ (A ∩ 𝐶) = (𝐶 ∩ A) | |
4 | inv1 3247 | . . . 4 ⊢ (B ∩ V) = B | |
5 | 3, 4 | xpeq12i 4310 | . . 3 ⊢ ((A ∩ 𝐶) × (B ∩ V)) = ((𝐶 ∩ A) × B) |
6 | 1, 2, 5 | 3eqtri 2061 | . 2 ⊢ ((A × B) ↾ 𝐶) = ((𝐶 ∩ A) × B) |
7 | df-ss 2925 | . . . 4 ⊢ (𝐶 ⊆ A ↔ (𝐶 ∩ A) = 𝐶) | |
8 | 7 | biimpi 113 | . . 3 ⊢ (𝐶 ⊆ A → (𝐶 ∩ A) = 𝐶) |
9 | 8 | xpeq1d 4311 | . 2 ⊢ (𝐶 ⊆ A → ((𝐶 ∩ A) × B) = (𝐶 × B)) |
10 | 6, 9 | syl5eq 2081 | 1 ⊢ (𝐶 ⊆ A → ((A × B) ↾ 𝐶) = (𝐶 × B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 Vcvv 2551 ∩ cin 2910 ⊆ wss 2911 × cxp 4286 ↾ cres 4290 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 df-rel 4295 df-res 4300 |
This theorem is referenced by: (None) |
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