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Theorem xpssres 6037
Description: Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
xpssres (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Proof of Theorem xpssres
StepHypRef Expression
1 df-res 5700 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 5844 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 inv1 4403 . . . 4 (𝐵 ∩ V) = 𝐵
43xpeq2i 5715 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
51, 2, 43eqtri 2766 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴𝐶) × 𝐵)
6 sseqin2 4230 . . . 4 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
76biimpi 216 . . 3 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
87xpeq1d 5717 . 2 (𝐶𝐴 → ((𝐴𝐶) × 𝐵) = (𝐶 × 𝐵))
95, 8eqtrid 2786 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  Vcvv 3477  cin 3961  wss 3962   × cxp 5686  cres 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-xp 5694  df-rel 5695  df-res 5700
This theorem is referenced by:  fparlem3  8137  fparlem4  8138  fpwwe2lem12  10679  pwssplit3  21077  cnconst2  23306  xkoccn  23642  tmdgsum  24118  dvcmul  25995  dvcmulf  25996  ply1gsumz  33598  lbsdiflsp0  33653  dvsconst  44325  dvsid  44326  aacllem  49031
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