Theorem xpssres 5992

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 5653	. . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2		inxp 5798	. . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
3		inv1 4364	. . . 4 ⊢ (𝐵 ∩ V) = 𝐵
4	3	xpeq2i 5668	. . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
5	1, 2, 4	3eqtri 2757	. 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵)
6		sseqin2 4189	. . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
7	6	biimpi 216	. . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
8	7	xpeq1d 5670	. 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵))
9	5, 8	eqtrid 2777	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917   × cxp 5639   ↾ cres 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648  df-res 5653

This theorem is referenced by:  fparlem3  8096  fparlem4  8097  fpwwe2lem12  10602  pwssplit3  20975  cnconst2  23177  xkoccn  23513  tmdgsum  23989  dvcmul  25854  dvcmulf  25855  ply1gsumz  33571  lbsdiflsp0  33629  dvsconst  44326  dvsid  44327  aacllem  49794