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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelsym | Structured version Visualization version GIF version | ||
| Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| eqvrelsym.1 | ⊢ (𝜑 → EqvRel 𝑅) | 
| eqvrelsym.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) | 
| Ref | Expression | 
|---|---|
| eqvrelsym | ⊢ (𝜑 → 𝐵𝑅𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqvrelsym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
| 2 | eqvrelsym.1 | . . . 4 ⊢ (𝜑 → EqvRel 𝑅) | |
| 3 | eqvrelrel 38599 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
| 4 | relbrcnvg 6122 | . . . 4 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
| 5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | 
| 6 | 1, 5 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) | 
| 7 | eqvrelsymrel 38601 | . . . 4 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) | |
| 8 | dfsymrel2 38551 | . . . . 5 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
| 9 | 8 | simplbi 497 | . . . 4 ⊢ ( SymRel 𝑅 → ◡𝑅 ⊆ 𝑅) | 
| 10 | 2, 7, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) | 
| 11 | 10 | ssbrd 5185 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) | 
| 12 | 6, 11 | mpd 15 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ⊆ wss 3950 class class class wbr 5142 ◡ccnv 5683 Rel wrel 5689 SymRel wsymrel 38195 EqvRel weqvrel 38200 | 
| 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-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-refrel 38514 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 | 
| This theorem is referenced by: eqvrelsymb 38608 eqvreltr4d 38611 eqvrelth 38613 | 
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