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Theorem elres 5470
Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
Assertion
Ref Expression
elres (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elres
StepHypRef Expression
1 relres 5461 . . . . 5 Rel (𝐵𝐶)
2 elrel 5256 . . . . 5 ((Rel (𝐵𝐶) ∧ 𝐴 ∈ (𝐵𝐶)) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
31, 2mpan 706 . . . 4 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4 eleq1 2718 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
54biimpd 219 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
6 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
76opelres 5436 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
87biimpi 206 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
98ancomd 466 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
105, 9syl6com 37 . . . . . . 7 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1110ancld 575 . . . . . 6 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12 an12 855 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1311, 12syl6ib 241 . . . . 5 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14132eximdv 1888 . . . 4 (𝐴 ∈ (𝐵𝐶) → (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
153, 14mpd 15 . . 3 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
16 rexcom4 3256 . . . 4 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17 df-rex 2947 . . . . 5 (∃𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817exbii 1814 . . . 4 (∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19 excom 2082 . . . 4 (∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2016, 18, 193bitri 286 . . 3 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2115, 20sylibr 224 . 2 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
227simplbi2com 656 . . . . . 6 (𝑥𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
234biimprd 238 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → 𝐴 ∈ (𝐵𝐶)))
2422, 23syl9 77 . . . . 5 (𝑥𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝐴 ∈ (𝐵𝐶))))
2524impd 446 . . . 4 (𝑥𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2625exlimdv 1901 . . 3 (𝑥𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2726rexlimiv 3056 . 2 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶))
2821, 27impbii 199 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cop 4216  cres 5145  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  elsnres  5471  eldm3  31777
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