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Theorem xpssres 5643
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 5324 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 5458 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 inv1 4166 . . . 4 (𝐵 ∩ V) = 𝐵
43xpeq2i 5339 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐴𝐶) × 𝐵)
51, 2, 43eqtri 2825 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴𝐶) × 𝐵)
6 sseqin2 4015 . . . 4 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
76biimpi 208 . . 3 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
87xpeq1d 5341 . 2 (𝐶𝐴 → ((𝐴𝐶) × 𝐵) = (𝐶 × 𝐵))
95, 8syl5eq 2845 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  Vcvv 3385  cin 3768  wss 3769   × cxp 5310  cres 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318  df-rel 5319  df-res 5324
This theorem is referenced by:  fparlem3  7516  fparlem4  7517  fpwwe2lem13  9752  pwssplit3  19382  cnconst2  21416  xkoccn  21751  tmdgsum  22227  dvcmul  24048  dvcmulf  24049  dvsconst  39311  dvsid  39312  aacllem  43349
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