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Theorem elinxp 5611
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 5408 . . . . 5 Rel (𝑅 ∩ (𝐴 × 𝐵))
2 elrel 5393 . . . . 5 ((Rel (𝑅 ∩ (𝐴 × 𝐵)) ∧ 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
31, 2mpan 681 . . . 4 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
4 eleq1 2832 . . . . . . . . 9 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
54biimpd 220 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
6 opelinxp 5353 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
76biimpi 207 . . . . . . . 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 546 . . . . . 6 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
10 an12 635 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
119, 10syl6ib 242 . . . . 5 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
12112eximdv 2014 . . . 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 3208 . . 3 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
1513, 14sylibr 225 . 2 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
166simplbi2 494 . . . . 5 ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
174biimprd 239 . . . . 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 398 . . 3 ((𝑥𝐴𝑦𝐵) → ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
2019rexlimivv 3183 . 2 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))
2115, 20impbii 200 1 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  cin 3733  cop 4342   × cxp 5277  Rel wrel 5284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286
This theorem is referenced by:  elres  5612  elidinxp  5634
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