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Theorem bj-opelidres 36346
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 36347 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 36345 . . 3 ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
21eleq2i 2824 . 2 (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3 elin 3965 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4 inex1g 5320 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 bj-opelid 36341 . . . . . 6 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
64, 5syl 17 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7 opelxp 5713 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉))
87a1i 11 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴𝑉𝐵𝑉)))
96, 8anbi12d 630 . . . 4 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
10 simpl 482 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 = 𝐵)
11 eleq1 2820 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1211biimpcd 248 . . . . . . 7 (𝐴𝑉 → (𝐴 = 𝐵𝐵𝑉))
1312anc2li 555 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉)))
1413ancld 550 . . . . 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 395   = wceq 1540  wcel 2105  Vcvv 3473  cin 3948  cop 4635   I cid 5574   × cxp 5675  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by: (None)
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