Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elinxp Structured version   Visualization version   GIF version

Theorem elinxp 5866
 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 5661 . . . . 5 Rel (𝑅 ∩ (𝐴 × 𝐵))
2 elrel 5645 . . . . 5 ((Rel (𝑅 ∩ (𝐴 × 𝐵)) ∧ 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
31, 2mpan 689 . . . 4 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
4 eleq1 2839 . . . . . . . . 9 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
54biimpd 232 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
6 opelinxp 5605 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
76biimpi 219 . . . . . . . 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 554 . . . . . 6 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
10 an12 644 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
119, 10syl6ib 254 . . . . 5 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
12112eximdv 1920 . . . 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 3227 . . 3 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
1513, 14sylibr 237 . 2 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
166simplbi2 504 . . . . 5 ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
174biimprd 251 . . . . 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 414 . . 3 ((𝑥𝐴𝑦𝐵) → ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
2019rexlimivv 3216 . 2 (∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))
2115, 20impbii 212 1 (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859  ⟨cop 4531   × cxp 5526  Rel wrel 5533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535 This theorem is referenced by:  elres  5867  elidinxp  5888
 Copyright terms: Public domain W3C validator