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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 35847 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | relbrcnvg 5968 | . . . 4 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
6 | 1, 5 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) |
7 | eqvrelsymrel 35849 | . . . 4 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) | |
8 | dfsymrel2 35800 | . . . . 5 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
9 | 8 | simplbi 500 | . . . 4 ⊢ ( SymRel 𝑅 → ◡𝑅 ⊆ 𝑅) |
10 | 2, 7, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
11 | 10 | ssbrd 5109 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
12 | 6, 11 | mpd 15 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ⊆ wss 3936 class class class wbr 5066 ◡ccnv 5554 Rel wrel 5560 SymRel wsymrel 35480 EqvRel weqvrel 35485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-refrel 35767 df-symrel 35795 df-trrel 35825 df-eqvrel 35835 |
This theorem is referenced by: eqvrelsymb 35856 eqvreltr4d 35859 eqvrelth 35861 |
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