Theorem xpssres 5048

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 4737	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 4864	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		incom 3399	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		inv1 3531	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
5	3, 4	xpeq12i 4747	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐶 ∩ 𝐴) × 𝐵)
6	1, 2, 5	3eqtri 2256	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶 ∩ 𝐴) × 𝐵)
7		df-ss 3213	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
8	7	biimpi 120	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶)
9	8	xpeq1d 4748	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐶 ∩ 𝐴) × 𝐵) = (𝐶 × 𝐵))
10	6, 9	eqtrid 2276	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Syntax hints:   → wi 4   = wceq 1397  Vcvv 2802   ∩ cin 3199   ⊆ wss 3200   × cxp 4723   ↾ cres 4727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  cnconst2  14963