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Theorem xpima1 5093
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 4657 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4656 . . . 4 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4873 . . 3 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4779 . . . 4 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4873 . . 3 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2214 . 2 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 xpeq1 4658 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8 0xp 4724 . . . 4 (∅ × (𝐵 ∩ V)) = ∅
97, 8eqtrdi 2238 . . 3 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
10 rneq 4872 . . . 4 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ∅)
11 rn0 4901 . . . 4 ran ∅ = ∅
1210, 11eqtrdi 2238 . . 3 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
139, 12syl 14 . 2 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
146, 13eqtrid 2234 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  Vcvv 2752  cin 3143  c0 3437   × cxp 4642  ran crn 4645  cres 4646  cima 4647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657
This theorem is referenced by: (None)
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