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Theorem xpssres 5422
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 5116 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 5243 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 inv1 3961 . . . 4 (𝐵 ∩ V) = 𝐵
43xpeq2i 5126 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
51, 2, 43eqtri 2646 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴𝐶) × 𝐵)
6 sseqin2 3809 . . . 4 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
76biimpi 206 . . 3 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
87xpeq1d 5128 . 2 (𝐶𝐴 → ((𝐴𝐶) × 𝐵) = (𝐶 × 𝐵))
95, 8syl5eq 2666 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  Vcvv 3195  cin 3566  wss 3567   × cxp 5102  cres 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110  df-rel 5111  df-res 5116
This theorem is referenced by:  fparlem3  7264  fparlem4  7265  fpwwe2lem13  9449  pwssplit3  19042  cnconst2  21068  xkoccn  21403  tmdgsum  21880  dvcmul  23688  dvcmulf  23689  dvsconst  38349  dvsid  38350  aacllem  42312
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