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Theorem elres 4902
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 4894 . . . . 5 Rel (𝐵𝐶)
2 elrel 4688 . . . . 5 ((Rel (𝐵𝐶) ∧ 𝐴 ∈ (𝐵𝐶)) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
31, 2mpan 421 . . . 4 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4 eleq1 2220 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
54biimpd 143 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
6 vex 2715 . . . . . . . . . . 11 𝑦 ∈ V
76opelres 4871 . . . . . . . . . 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 551 . . . . . 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 1862 . . . 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 2735 . . . 4 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17 df-rex 2441 . . . . 5 (∃𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817exbii 1585 . . . 4 (∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19 excom 1644 . . . 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 1424 . . . . . 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 1799 . . 3 (𝑥𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2726rexlimiv 2568 . 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 1335  wex 1472  wcel 2128  wrex 2436  cop 3563  cres 4588  Rel wrel 4591
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592  df-rel 4593  df-res 4598
This theorem is referenced by:  elsnres  4903
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