Theorem xpssres 4954

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 4650	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 4773	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		incom 3339	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		inv1 3471	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
5	3, 4	xpeq12i 4660	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐶 ∩ 𝐴) × 𝐵)
6	1, 2, 5	3eqtri 2212	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶 ∩ 𝐴) × 𝐵)
7		df-ss 3154	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
8	7	biimpi 120	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶)
9	8	xpeq1d 4661	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐶 ∩ 𝐴) × 𝐵) = (𝐶 × 𝐵))
10	6, 9	eqtrid 2232	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Syntax hints:   → wi 4   = wceq 1363  Vcvv 2749   ∩ cin 3140   ⊆ wss 3141   × cxp 4636   ↾ cres 4640

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-xp 4644  df-rel 4645  df-res 4650

This theorem is referenced by:  cnconst2  14029