Theorem xpssres 4588

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 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))

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)