Theorem xpssres 5889

Description: Restriction of a constant function (or other Cartesian 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 5567	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 5703	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		inv1 4348	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
4	3	xpeq2i 5582	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
5	1, 2, 4	3eqtri 2848	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵)
6		sseqin2 4192	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
7	6	biimpi 218	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
8	7	xpeq1d 5584	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵))
9	5, 8	syl5eq 2868	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   × cxp 5553   ↾ cres 5557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561  df-rel 5562  df-res 5567

This theorem is referenced by:  fparlem3  7809  fparlem4  7810  fpwwe2lem13  10064  pwssplit3  19833  cnconst2  21891  xkoccn  22227  tmdgsum  22703  dvcmul  24541  dvcmulf  24542  lbsdiflsp0  31022  dvsconst  40682  dvsid  40683  aacllem  44922