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Theorem bj-elid 32093
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 5063 . . . . 5 Rel I
2 df-rel 4939 . . . . 5 (Rel I ↔ I ⊆ (V × V))
31, 2mpbi 218 . . . 4 I ⊆ (V × V)
43sseli 3468 . . 3 (𝐴 ∈ I → 𝐴 ∈ (V × V))
5 1st2nd2 6971 . . . . . . 7 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
64, 5syl 17 . . . . . 6 (𝐴 ∈ I → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76eleq1d 2576 . . . . 5 (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
87ibi 254 . . . 4 (𝐴 ∈ I → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I )
9 df-id 4847 . . . . . 6 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
109eleq2i 2584 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11 fvex 5997 . . . . . 6 (1st𝐴) ∈ V
12 fvex 5997 . . . . . 6 (2nd𝐴) ∈ V
13 eqeq12 2527 . . . . . 6 ((𝑥 = (1st𝐴) ∧ 𝑦 = (2nd𝐴)) → (𝑥 = 𝑦 ↔ (1st𝐴) = (2nd𝐴)))
1411, 12, 13opelopaba 4810 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st𝐴) = (2nd𝐴))
1510, 14bitri 262 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴))
168, 15sylib 206 . . 3 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
174, 16jca 552 . 2 (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
185eleq1d 2576 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
1918biimprd 236 . . . 4 (𝐴 ∈ (V × V) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I → 𝐴 ∈ I ))
2015, 19syl5bir 231 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
2120imp 443 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)) → 𝐴 ∈ I )
2217, 21impbii 197 1 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  wss 3444  cop 4034  {copab 4540   I cid 4842   × cxp 4930  Rel wrel 4937  cfv 5689  1st c1st 6932  2nd c2nd 6933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-1st 6934  df-2nd 6935
This theorem is referenced by:  bj-elid2  32094  bj-elid3  32095
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