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Mirrors > Home > MPE Home > Th. List > elid | Structured version Visualization version GIF version |
Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6209 after elrid 6055. (Contributed by BJ, 28-Aug-2022.) |
Ref | Expression |
---|---|
elid | ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5832 | . . . . 5 ⊢ Rel I | |
2 | dfrel3 6209 | . . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
3 | 1, 2 | mpbi 229 | . . . 4 ⊢ ( I ↾ V) = I |
4 | 3 | eqcomi 2735 | . . 3 ⊢ I = ( I ↾ V) |
5 | 4 | eleq2i 2818 | . 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V)) |
6 | elrid 6055 | . 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉) | |
7 | rexv 3490 | . 2 ⊢ (∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | |
8 | 5, 6, 7 | 3bitri 296 | 1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∃wrex 3060 Vcvv 3462 〈cop 4639 I cid 5579 ↾ cres 5684 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-res 5694 |
This theorem is referenced by: (None) |
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