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Theorem elxp2 4419
Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
Assertion
Ref Expression
elxp2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp2
StepHypRef Expression
1 df-rex 2359 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)))
2 r19.42v 2517 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3 an13 528 . . . . 5 ((𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
43exbii 1537 . . . 4 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
51, 2, 43bitr3i 208 . . 3 ((𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
65exbii 1537 . 2 (∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
7 df-rex 2359 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8 elxp 4418 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
96, 7, 83bitr4ri 211 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wrex 2354  cop 3425   × cxp 4399
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4407
This theorem is referenced by:  opelxp  4430  xpiundi  4454  xpiundir  4455  ssrel2  4486  f1o2ndf1  5928  xpdom2  6477  elreal  7269
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