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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 5833 after elrid 5695. (Contributed by BJ, 28-Aug-2022.) |
Ref | Expression |
---|---|
elid | ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5483 | . . . . 5 ⊢ Rel I | |
2 | dfrel3 5833 | . . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
3 | 1, 2 | mpbi 222 | . . . 4 ⊢ ( I ↾ V) = I |
4 | 3 | eqcomi 2835 | . . 3 ⊢ I = ( I ↾ V) |
5 | 4 | eleq2i 2899 | . 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V)) |
6 | elrid 5695 | . 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉) | |
7 | rexv 3438 | . 2 ⊢ (∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | |
8 | 5, 6, 7 | 3bitri 289 | 1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ∃wrex 3119 Vcvv 3415 〈cop 4404 I cid 5250 ↾ cres 5345 Rel wrel 5348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-res 5355 |
This theorem is referenced by: (None) |
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