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| Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6217 after elrid 6063. (Contributed by BJ, 28-Aug-2022.) | 
| Ref | Expression | 
|---|---|
| elid | ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reli 5835 | . . . . 5 ⊢ Rel I | |
| 2 | dfrel3 6217 | . . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
| 3 | 1, 2 | mpbi 230 | . . . 4 ⊢ ( I ↾ V) = I | 
| 4 | 3 | eqcomi 2745 | . . 3 ⊢ I = ( I ↾ V) | 
| 5 | 4 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V)) | 
| 6 | elrid 6063 | . 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉) | |
| 7 | rexv 3508 | . 2 ⊢ (∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | |
| 8 | 5, 6, 7 | 3bitri 297 | 1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 〈cop 4631 I cid 5576 ↾ cres 5686 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-res 5696 | 
| This theorem is referenced by: (None) | 
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