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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 38579 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | relbrcnvg 6126 | . . . 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 38581 | . . . 4 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) | |
8 | dfsymrel2 38531 | . . . . 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 5191 | . 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 3963 class class class wbr 5148 ◡ccnv 5688 Rel wrel 5694 SymRel wsymrel 38174 EqvRel weqvrel 38179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 |
This theorem is referenced by: eqvrelsymb 38588 eqvreltr4d 38591 eqvrelth 38593 |
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