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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 6057 after elrid 5915. (Contributed by BJ, 28-Aug-2022.) |
Ref | Expression |
---|---|
elid | ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5700 | . . . . 5 ⊢ Rel I | |
2 | dfrel3 6057 | . . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
3 | 1, 2 | mpbi 232 | . . . 4 ⊢ ( I ↾ V) = I |
4 | 3 | eqcomi 2832 | . . 3 ⊢ I = ( I ↾ V) |
5 | 4 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V)) |
6 | elrid 5915 | . 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉) | |
7 | rexv 3522 | . 2 ⊢ (∃𝑥 ∈ V 𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) | |
8 | 5, 6, 7 | 3bitri 299 | 1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 〈cop 4575 I cid 5461 ↾ cres 5559 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-res 5569 |
This theorem is referenced by: (None) |
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