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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 36303 | . . 3 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | |
2 | cnvsym 5968 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
3 | 1, 2 | bitr4di 292 | . 2 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅)) |
4 | eqcom 2741 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅) | |
5 | 3, 4 | bitr3di 289 | 1 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 ◡ccnv 5539 Rels crels 36029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-rels 36297 |
This theorem is referenced by: elrelscnveq4 36306 dfsymrels4 36355 |
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