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Theorem xpssres 6036
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 5697 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 5842 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 inv1 4398 . . . 4 (𝐵 ∩ V) = 𝐵
43xpeq2i 5712 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
51, 2, 43eqtri 2769 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴𝐶) × 𝐵)
6 sseqin2 4223 . . . 4 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
76biimpi 216 . . 3 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
87xpeq1d 5714 . 2 (𝐶𝐴 → ((𝐴𝐶) × 𝐵) = (𝐶 × 𝐵))
95, 8eqtrid 2789 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Vcvv 3480  cin 3950  wss 3951   × cxp 5683  cres 5687
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  fparlem3  8139  fparlem4  8140  fpwwe2lem12  10682  pwssplit3  21060  cnconst2  23291  xkoccn  23627  tmdgsum  24103  dvcmul  25981  dvcmulf  25982  ply1gsumz  33619  lbsdiflsp0  33677  dvsconst  44349  dvsid  44350  aacllem  49320
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