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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 6154 after elrid 6003. (Contributed by BJ, 28-Aug-2022.) |
Ref | Expression |
---|---|
elid | ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5786 | . . . . 5 ⊢ Rel I | |
2 | dfrel3 6154 | . . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
3 | 1, 2 | mpbi 229 | . . . 4 ⊢ ( I ↾ V) = I |
4 | 3 | eqcomi 2742 | . . 3 ⊢ I = ( I ↾ V) |
5 | 4 | eleq2i 2826 | . 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V)) |
6 | elrid 6003 | . 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩) | |
7 | rexv 3472 | . 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 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 Vcvv 3447 ⟨cop 4596 I cid 5534 ↾ cres 5639 Rel wrel 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-res 5649 |
This theorem is referenced by: (None) |
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