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Theorem bj-elid4 36540
Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid4 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Proof of Theorem bj-elid4
StepHypRef Expression
1 1st2nd2 8008 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 eleq1 2813 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
32adantl 481 . . 3 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
4 fvex 6895 . . . . 5 (2nd𝐴) ∈ V
54inex2 5309 . . . 4 ((1st𝐴) ∩ (2nd𝐴)) ∈ V
6 bj-opelid 36528 . . . 4 (((1st𝐴) ∩ (2nd𝐴)) ∈ V → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴)))
75, 6mp1i 13 . . 3 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴)))
83, 7bitrd 279 . 2 ((𝐴 ∈ (𝑉 × 𝑊) ∧ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
91, 8mpdan 684 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cin 3940  cop 4627   I cid 5564   × cxp 5665  cfv 6534  1st c1st 7967  2nd c2nd 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-1st 7969  df-2nd 7970
This theorem is referenced by:  bj-elid5  36541  bj-elid6  36542  bj-eldiag  36548
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