Theorem xpssres 6022

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 5678	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 5823	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		inv1 4358	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
4	3	xpeq2i 5693	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
5	1, 2, 4	3eqtri 2793	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵)
6		sseqin2 4179	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
7	6	biimpi 219	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
8	7	xpeq1d 5695	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵))
9	5, 8	eqtrid 2813	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908   × cxp 5664   ↾ cres 5668

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5672  df-rel 5673  df-res 5678

This theorem is used by:  fparlem3  8118  fparlem4  8119  fpwwe2lem12  10645  pwssplit3  21219  cnconst2  23477  xkoccn  23813  tmdgsum  24289  dvcmul  26140  dvcmulf  26141  ply1gsumz  33920  lbsdiflsp0  34047  dvsconst  45081  dvsid  45082  aacllem  50662