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Theorem elres 4927
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 4919 . . . . 5 Rel (𝐵𝐶)
2 elrel 4713 . . . . 5 ((Rel (𝐵𝐶) ∧ 𝐴 ∈ (𝐵𝐶)) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
31, 2mpan 422 . . . 4 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4 eleq1 2233 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
54biimpd 143 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
6 vex 2733 . . . . . . . . . . 11 𝑦 ∈ V
76opelres 4896 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
87biimpi 119 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
98ancomd 265 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
105, 9syl6com 35 . . . . . . 7 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1110ancld 323 . . . . . 6 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12 an12 556 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1311, 12syl6ib 160 . . . . 5 (𝐴 ∈ (𝐵𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14132eximdv 1875 . . . 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 2753 . . . 4 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17 df-rex 2454 . . . . 5 (∃𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817exbii 1598 . . . 4 (∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19 excom 1657 . . . 4 (∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2016, 18, 193bitri 205 . . 3 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥𝑦(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2115, 20sylibr 133 . 2 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
227simplbi2com 1437 . . . . . 6 (𝑥𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
234biimprd 157 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → 𝐴 ∈ (𝐵𝐶)))
2422, 23syl9 72 . . . . 5 (𝑥𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝐴 ∈ (𝐵𝐶))))
2524impd 252 . . . 4 (𝑥𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2625exlimdv 1812 . . 3 (𝑥𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2726rexlimiv 2581 . 2 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶))
2821, 27impbii 125 1 (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  wrex 2449  cop 3586  cres 4613  Rel wrel 4616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623
This theorem is referenced by:  elsnres  4928
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