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Theorem bj-elid 33663
 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 5495 . . . . 5 Rel I
2 df-rel 5362 . . . . 5 (Rel I ↔ I ⊆ (V × V))
31, 2mpbi 222 . . . 4 I ⊆ (V × V)
43sseli 3817 . . 3 (𝐴 ∈ I → 𝐴 ∈ (V × V))
5 1st2nd2 7484 . . . . . . 7 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
64, 5syl 17 . . . . . 6 (𝐴 ∈ I → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
76eleq1d 2844 . . . . 5 (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
87ibi 259 . . . 4 (𝐴 ∈ I → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I )
9 df-id 5261 . . . . . 6 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
109eleq2i 2851 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11 fvex 6459 . . . . . 6 (1st𝐴) ∈ V
12 fvex 6459 . . . . . 6 (2nd𝐴) ∈ V
13 eqeq12 2791 . . . . . 6 ((𝑥 = (1st𝐴) ∧ 𝑦 = (2nd𝐴)) → (𝑥 = 𝑦 ↔ (1st𝐴) = (2nd𝐴)))
1411, 12, 13opelopaba 5228 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st𝐴) = (2nd𝐴))
1510, 14bitri 267 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ↔ (1st𝐴) = (2nd𝐴))
168, 15sylib 210 . . 3 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
174, 16jca 507 . 2 (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
185eleq1d 2844 . . . . 5 (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ I ))
1918biimprd 240 . . . 4 (𝐴 ∈ (V × V) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ I → 𝐴 ∈ I ))
2015, 19syl5bir 235 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
2120imp 397 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)) → 𝐴 ∈ I )
2217, 21impbii 201 1 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  ⟨cop 4404  {copab 4948   I cid 5260   × cxp 5353  Rel wrel 5360  ‘cfv 6135  1st c1st 7443  2nd c2nd 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-1st 7445  df-2nd 7446 This theorem is referenced by:  bj-elid2  33664  bj-elid3  33665
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