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Theorem elres 4646
 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 4639 . . . . 5 Rel (𝐵𝐶)
2 elrel 4442 . . . . 5 ((Rel (𝐵𝐶) ∧ 𝐴 ∈ (𝐵𝐶)) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
31, 2mpan 400 . . . 4 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4 eleq1 2100 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
54biimpd 132 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
6 vex 2560 . . . . . . . . . . 11 𝑦 ∈ V
76opelres 4617 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
87biimpi 113 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
98ancomd 254 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
105, 9syl6com 31 . . . . . . 7 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1110ancld 308 . . . . . 6 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12 an12 495 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1311, 12syl6ib 150 . . . . 5 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14132eximdv 1762 . . . 4 (𝐴 ∈ (𝐵𝐶) → (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
153, 14mpd 13 . . 3 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
16 rexcom4 2577 . . . 4 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17 df-rex 2312 . . . . 5 (∃𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817exbii 1496 . . . 4 (∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19 excom 1554 . . . 4 (∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2016, 18, 193bitri 195 . . 3 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2115, 20sylibr 137 . 2 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
227simplbi2com 1333 . . . . . 6 (𝑥𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
234biimprd 147 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → 𝐴 ∈ (𝐵𝐶)))
2422, 23syl9 66 . . . . 5 (𝑥𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝐴 ∈ (𝐵𝐶))))
2524impd 242 . . . 4 (𝑥𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2625exlimdv 1700 . . 3 (𝑥𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2726rexlimiv 2427 . 2 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶))
2821, 27impbii 117 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   ↾ cres 4347  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  elsnres  4647
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