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Theorem bj-opelidres 34882
 Description: Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 34883 from it. (Contributed by BJ, 29-Mar-2020.)
Assertion
Ref Expression
bj-opelidres (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Proof of Theorem bj-opelidres
StepHypRef Expression
1 bj-idres 34881 . . 3 ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
21eleq2i 2843 . 2 (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3 elin 3876 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4 inex1g 5192 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 bj-opelid 34877 . . . . . 6 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
64, 5syl 17 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7 opelxp 5563 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉))
87a1i 11 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉)))
96, 8anbi12d 633 . . . 4 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
10 simpl 486 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 = 𝐵)
11 eleq1 2839 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1211biimpcd 252 . . . . . . 7 (𝐴𝑉 → (𝐴 = 𝐵𝐵𝑉))
1312anc2li 559 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉)))
1413ancld 554 . . . . 5 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
1510, 14impbid2 229 . . . 4 (𝐴𝑉 → ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) ↔ 𝐴 = 𝐵))
169, 15bitrd 282 . . 3 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
173, 16syl5bb 286 . 2 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
182, 17syl5bb 286 1 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859  ⟨cop 4531   I cid 5432   × cxp 5525   ↾ cres 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-res 5539 This theorem is referenced by: (None)
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