Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-opelidres Structured version   Visualization version   GIF version

Theorem bj-opelidres 37129
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 37130 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 37128 . . 3 ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
21eleq2i 2836 . 2 (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3 elin 3992 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4 inex1g 5337 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 bj-opelid 37124 . . . . . 6 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
64, 5syl 17 . . . . 5 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7 opelxp 5736 . . . . . 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 482 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) → 𝐴 = 𝐵)
11 eleq1 2832 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1211biimpcd 249 . . . . . . 7 (𝐴𝑉 → (𝐴 = 𝐵𝐵𝑉))
1312anc2li 555 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉)))
1413ancld 550 . . . . 5 (𝐴𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉))))
1510, 14impbid2 226 . . . 4 (𝐴𝑉 → ((𝐴 = 𝐵 ∧ (𝐴𝑉𝐵𝑉)) ↔ 𝐴 = 𝐵))
169, 15bitrd 279 . . 3 (𝐴𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
173, 16bitrid 283 . 2 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
182, 17bitrid 283 1 (𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cin 3975  cop 4654   I cid 5592   × cxp 5698  cres 5702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator