Theorem xpssres 6015

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 5671	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 5816	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		inv1 4351	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
4	3	xpeq2i 5686	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
5	1, 2, 4	3eqtri 2789	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵)
6		sseqin2 4172	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
7	6	biimpi 219	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
8	7	xpeq1d 5688	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵))
9	5, 8	eqtrid 2809	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902   × cxp 5657   ↾ cres 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-xp 5665  df-rel 5666  df-res 5671

This theorem is used by:  fparlem3  8115  fparlem4  8116  fpwwe2lem12  10655  pwssplit3  21251  cnconst2  23514  xkoccn  23851  tmdgsum  24327  dvcmul  26178  dvcmulf  26179  ply1gsumz  34017  lbsdiflsp0  34144  dvsconst  45162  dvsid  45163  aacllem  50780