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Theorem xpima2m 5030
 Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2m (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem xpima2m
StepHypRef Expression
1 df-ima 4596 . . . 4 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4595 . . . . 5 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4811 . . . 4 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4717 . . . . 5 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4811 . . . 4 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2182 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 rnxpm 5012 . . 3 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ran ((𝐴𝐶) × (𝐵 ∩ V)) = (𝐵 ∩ V))
86, 7syl5eq 2202 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = (𝐵 ∩ V))
9 inv1 3430 . 2 (𝐵 ∩ V) = 𝐵
108, 9eqtrdi 2206 1 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335  ∃wex 1472   ∈ wcel 2128  Vcvv 2712   ∩ cin 3101   × cxp 4581  ran crn 4584   ↾ cres 4585   “ cima 4586 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590  df-cnv 4591  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596 This theorem is referenced by:  xpimasn  5031
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