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Theorem xpssres 4924
Description: Restriction of a constant function (or other cross 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 4621 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 4743 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 incom 3319 . . . 4 (𝐴𝐶) = (𝐶𝐴)
4 inv1 3450 . . . 4 (𝐵 ∩ V) = 𝐵
53, 4xpeq12i 4631 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐶𝐴) × 𝐵)
61, 2, 53eqtri 2195 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶𝐴) × 𝐵)
7 df-ss 3134 . . . 4 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
87biimpi 119 . . 3 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
98xpeq1d 4632 . 2 (𝐶𝐴 → ((𝐶𝐴) × 𝐵) = (𝐶 × 𝐵))
106, 9eqtrid 2215 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  Vcvv 2730  cin 3120  wss 3121   × cxp 4607  cres 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-xp 4615  df-rel 4616  df-res 4621
This theorem is referenced by:  cnconst2  12986
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