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Theorem elres 5399
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 5390 . . . . 5 Rel (𝐵𝐶)
2 elrel 5188 . . . . 5 ((Rel (𝐵𝐶) ∧ 𝐴 ∈ (𝐵𝐶)) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
31, 2mpan 705 . . . 4 (𝐴 ∈ (𝐵𝐶) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4 eleq1 2686 . . . . . . . . 9 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
54biimpd 219 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶)))
6 vex 3192 . . . . . . . . . . 11 𝑦 ∈ V
76opelres 5366 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
87biimpi 206 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐵𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵𝑥𝐶))
98ancomd 467 . . . . . . . 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 837 . . . . . 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 1845 . . . 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 3214 . . . 4 (∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17 df-rex 2913 . . . . 5 (∃𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817exbii 1771 . . . 4 (∃𝑦𝑥𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥(𝑥𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19 excom 2039 . . . 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 447 . . . 4 (𝑥𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2625exlimdv 1858 . . 3 (𝑥𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵𝐶)))
2726rexlimiv 3021 . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4159  cres 5081  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-opab 4679  df-xp 5085  df-rel 5086  df-res 5091
This theorem is referenced by:  elsnres  5400  eldm3  31395
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