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Theorem xpima1 4953
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 4520 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4519 . . . 4 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4735 . . 3 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4641 . . . 4 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4735 . . 3 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2140 . 2 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 xpeq1 4521 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8 0xp 4587 . . . 4 (∅ × (𝐵 ∩ V)) = ∅
97, 8syl6eq 2164 . . 3 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
10 rneq 4734 . . . 4 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ∅)
11 rn0 4763 . . . 4 ran ∅ = ∅
1210, 11syl6eq 2164 . . 3 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
139, 12syl 14 . 2 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
146, 13syl5eq 2160 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  Vcvv 2658  cin 3038  c0 3331   × cxp 4505  ran crn 4508  cres 4509  cima 4510
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by: (None)
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