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Theorem xpssres 4734
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 4440 . . 3 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
2 inxp 4558 . . 3 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
3 incom 3190 . . . 4 (𝐴𝐶) = (𝐶𝐴)
4 inv1 3316 . . . 4 (𝐵 ∩ V) = 𝐵
53, 4xpeq12i 4450 . . 3 ((𝐴𝐶) × (𝐵 ∩ V)) = ((𝐶𝐴) × 𝐵)
61, 2, 53eqtri 2112 . 2 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐶𝐴) × 𝐵)
7 df-ss 3010 . . . 4 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
87biimpi 118 . . 3 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
98xpeq1d 4451 . 2 (𝐶𝐴 → ((𝐶𝐴) × 𝐵) = (𝐶 × 𝐵))
106, 9syl5eq 2132 1 (𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  Vcvv 2619  cin 2996  wss 2997   × cxp 4426  cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by: (None)
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