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Theorem xpima1 4767
 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 4358 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4357 . . . 4 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4562 . . 3 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4470 . . . 4 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4562 . . 3 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2064 . 2 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 xpeq1 4359 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8 0xp 4420 . . . 4 (∅ × (𝐵 ∩ V)) = ∅
97, 8syl6eq 2088 . . 3 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
10 rneq 4561 . . . 4 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ∅)
11 rn0 4588 . . . 4 ran ∅ = ∅
1210, 11syl6eq 2088 . . 3 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
139, 12syl 14 . 2 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
146, 13syl5eq 2084 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  Vcvv 2557   ∩ cin 2916  ∅c0 3224   × cxp 4343  ran crn 4346   ↾ cres 4347   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
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