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Theorem bj-elid 32262
 Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Proof of Theorem bj-elid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5171 . . . . 5 Rel I
2 df-rel 5045 . . . . 5 (Rel I ↔ I ⊆ (V × V))
31, 2mpbi 219 . . . 4 I ⊆ (V × V)
43sseli 3564 . . 3 (𝐴 ∈ I → 𝐴 ∈ (V × V))
5 1st2nd2 7096 . . . . . . 7 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
64, 5syl 17 . . . . . 6 (𝐴 ∈ I → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76eleq1d 2672 . . . . 5 (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
87ibi 255 . . . 4 (𝐴 ∈ I → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I )
9 df-id 4953 . . . . . 6 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
109eleq2i 2680 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11 fvex 6113 . . . . . 6 (1st𝐴) ∈ V
12 fvex 6113 . . . . . 6 (2nd𝐴) ∈ V
13 eqeq12 2623 . . . . . 6 ((𝑥 = (1st𝐴) ∧ 𝑦 = (2nd𝐴)) → (𝑥 = 𝑦 ↔ (1st𝐴) = (2nd𝐴)))
1411, 12, 13opelopaba 4916 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st𝐴) = (2nd𝐴))
1510, 14bitri 263 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴))
168, 15sylib 207 . . 3 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
174, 16jca 553 . 2 (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
185eleq1d 2672 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
1918biimprd 237 . . . 4 (𝐴 ∈ (V × V) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I → 𝐴 ∈ I ))
2015, 19syl5bir 232 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
2120imp 444 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)) → 𝐴 ∈ I )
2217, 21impbii 198 1 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  {copab 4642   I cid 4948   × cxp 5036  Rel wrel 5043  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  bj-elid2  32263  bj-elid3  32264
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