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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 37980 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | relbrcnvg 6098 | . . . 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 37982 | . . . 4 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) | |
8 | dfsymrel2 37932 | . . . . 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 5184 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
12 | 6, 11 | mpd 15 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ⊆ wss 3943 class class class wbr 5141 ◡ccnv 5668 Rel wrel 5674 SymRel wsymrel 37568 EqvRel weqvrel 37573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-refrel 37895 df-symrel 37927 df-trrel 37957 df-eqvrel 37968 |
This theorem is referenced by: eqvrelsymb 37989 eqvreltr4d 37992 eqvrelth 37994 |
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