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Theorem xpssres 5870
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 5548 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 5684 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 inv1 4329 . . . 4 (𝐵 ∩ V) = 𝐵
43xpeq2i 5563 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
51, 2, 43eqtri 2851 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴𝐶) × 𝐵)
6 sseqin2 4175 . . . 4 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
76biimpi 219 . . 3 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
87xpeq1d 5565 . 2 (𝐶𝐴 → ((𝐴𝐶) × 𝐵) = (𝐶 × 𝐵))
95, 8syl5eq 2871 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  Vcvv 3479  cin 3917  wss 3918   × cxp 5534  cres 5538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-opab 5110  df-xp 5542  df-rel 5543  df-res 5548
This theorem is referenced by:  fparlem3  7792  fparlem4  7793  fpwwe2lem13  10049  pwssplit3  19819  cnconst2  21877  xkoccn  22213  tmdgsum  22689  dvcmul  24536  dvcmulf  24537  lbsdiflsp0  31043  dvsconst  40870  dvsid  40871  aacllem  45159
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