Theorem xpssres 4645

Description: Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
xpssres	⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Proof of Theorem xpssres

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))

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)