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Theorem elinxp 5888
Description: Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elinxp (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑅,𝑦

Proof of Theorem elinxp
StepHypRef Expression
1 relinxp 5685 . . . . 5 Rel (𝑅 ∩ (𝐴 × 𝐵))
2 elrel 5669 . . . . 5 ((Rel (𝑅 ∩ (𝐴 × 𝐵)) ∧ 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
31, 2mpan 686 . . . 4 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
4 eleq1 2904 . . . . . . . . 9 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
54biimpd 230 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
6 opelinxp 5629 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
76biimpi 217 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
85, 7syl6com 37 . . . . . . 7 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
98ancld 551 . . . . . 6 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
10 an12 641 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
119, 10syl6ib 252 . . . . 5 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
12112eximdv 1913 . . . 4 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
133, 12mpd 15 . . 3 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
14 r2ex 3307 . . 3 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
1513, 14sylibr 235 . 2 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
166simplbi2 501 . . . . 5 ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
174biimprd 249 . . . . 5 (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
1816, 17syl9 77 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))))
1918impd 411 . . 3 ((𝑥𝐴𝑦𝐵) → ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
2019rexlimivv 3296 . 2 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))
2115, 20impbii 210 1 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wrex 3143  cin 3938  cop 4569   × cxp 5551  Rel wrel 5558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560
This theorem is referenced by:  elres  5889  elidinxp  5909
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