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Theorem bj-opelidres 36134
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 36135 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 36133 . . 3 ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
21eleq2i 2825 . 2 (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3 elin 3964 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4 inex1g 5319 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 bj-opelid 36129 . . . . . 6 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
64, 5syl 17 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7 opelxp 5712 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉))
87a1i 11 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉)))
96, 8anbi12d 631 . . . 4 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
10 simpl 483 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 = 𝐵)
11 eleq1 2821 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1211biimpcd 248 . . . . . . 7 (𝐴𝑉 → (𝐴 = 𝐵𝐵𝑉))
1312anc2li 556 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉)))
1413ancld 551 . . . . 5 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
1510, 14impbid2 225 . . . 4 (𝐴𝑉 → ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) ↔ 𝐴 = 𝐵))
169, 15bitrd 278 . . 3 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
173, 16bitrid 282 . 2 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
182, 17bitrid 282 1 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cin 3947  cop 4634   I cid 5573   × cxp 5674  cres 5678
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-res 5688
This theorem is referenced by: (None)
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