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Theorem xpima1 4864
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Proof of Theorem xpima1
StepHypRef Expression
1 df-ima 4441 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4440 . . . 4 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4651 . . 3 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4558 . . . 4 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4651 . . 3 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2112 . 2 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 xpeq1 4442 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8 0xp 4506 . . . 4 (∅ × (𝐵 ∩ V)) = ∅
97, 8syl6eq 2136 . . 3 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
10 rneq 4650 . . . 4 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ∅)
11 rn0 4677 . . . 4 ran ∅ = ∅
1210, 11syl6eq 2136 . . 3 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
139, 12syl 14 . 2 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
146, 13syl5eq 2132 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  Vcvv 2619  cin 2996  c0 3284   × cxp 4426  ran crn 4429  cres 4430  cima 4431
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-in1 579  ax-in2 580  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-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by: (None)
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