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Theorem bj-elid 33215
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 5282 . . . . 5 Rel I
2 df-rel 5150 . . . . 5 (Rel I ↔ I ⊆ (V × V))
31, 2mpbi 220 . . . 4 I ⊆ (V × V)
43sseli 3632 . . 3 (𝐴 ∈ I → 𝐴 ∈ (V × V))
5 1st2nd2 7249 . . . . . . 7 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
64, 5syl 17 . . . . . 6 (𝐴 ∈ I → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76eleq1d 2715 . . . . 5 (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
87ibi 256 . . . 4 (𝐴 ∈ I → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I )
9 df-id 5053 . . . . . 6 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
109eleq2i 2722 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11 fvex 6239 . . . . . 6 (1st𝐴) ∈ V
12 fvex 6239 . . . . . 6 (2nd𝐴) ∈ V
13 eqeq12 2664 . . . . . 6 ((𝑥 = (1st𝐴) ∧ 𝑦 = (2nd𝐴)) → (𝑥 = 𝑦 ↔ (1st𝐴) = (2nd𝐴)))
1411, 12, 13opelopaba 5020 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st𝐴) = (2nd𝐴))
1510, 14bitri 264 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴))
168, 15sylib 208 . . 3 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
174, 16jca 553 . 2 (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
185eleq1d 2715 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
1918biimprd 238 . . . 4 (𝐴 ∈ (V × V) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I → 𝐴 ∈ I ))
2015, 19syl5bir 233 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
2120imp 444 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)) → 𝐴 ∈ I )
2217, 21impbii 199 1 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  cop 4216  {copab 4745   I cid 5052   × cxp 5141  Rel wrel 5148  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by:  bj-elid2  33216  bj-elid3  33217
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