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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrelscnveq | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
Ref | Expression |
---|---|
elrelscnveq | ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrelscnveq3 36597 | . . 3 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | |
2 | cnvsym 6017 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
3 | 1, 2 | bitr4di 289 | . 2 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅)) |
4 | eqcom 2747 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅) | |
5 | 3, 4 | bitr3di 286 | 1 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 class class class wbr 5079 ◡ccnv 5588 Rels crels 36323 |
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 2015 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5595 df-rel 5596 df-cnv 5597 df-rels 36591 |
This theorem is referenced by: elrelscnveq4 36600 dfsymrels4 36649 |
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