Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 36353 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | relbrcnvg 5942 | . . . 4 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
6 | 1, 5 | mpbird 260 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) |
7 | eqvrelsymrel 36355 | . . . 4 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) | |
8 | dfsymrel2 36306 | . . . . 5 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
9 | 8 | simplbi 501 | . . . 4 ⊢ ( SymRel 𝑅 → ◡𝑅 ⊆ 𝑅) |
10 | 2, 7, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
11 | 10 | ssbrd 5073 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
12 | 6, 11 | mpd 15 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ⊆ wss 3843 class class class wbr 5030 ◡ccnv 5524 Rel wrel 5530 SymRel wsymrel 35988 EqvRel weqvrel 35993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-refrel 36273 df-symrel 36301 df-trrel 36331 df-eqvrel 36341 |
This theorem is referenced by: eqvrelsymb 36362 eqvreltr4d 36365 eqvrelth 36367 |
Copyright terms: Public domain | W3C validator |