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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrelscnveq3 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
Ref | Expression |
---|---|
elrelscnveq3 | ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3960 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)) | |
2 | cnvsym 6067 | . . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
3 | 2 | biimpi 215 | . . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
4 | 3 | a1d 25 | . . . . 5 ⊢ (◡𝑅 ⊆ 𝑅 → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
5 | 4 | adantl 483 | . . . 4 ⊢ ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
6 | 5 | com12 32 | . . 3 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
7 | elrelsrelim 36996 | . . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅) | |
8 | dfrel2 6142 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
9 | 7, 8 | sylib 217 | . . . . 5 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅) |
10 | cnvss 5829 | . . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅) | |
11 | sseq1 3970 | . . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅)) | |
12 | 10, 11 | syl5ibcom 244 | . . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅)) |
13 | 2, 12 | sylbir 234 | . . . . 5 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅)) |
14 | 9, 13 | syl5com 31 | . . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅)) |
15 | 2 | biimpri 227 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅) |
16 | 14, 15 | jca2 515 | . . 3 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅))) |
17 | 6, 16 | impbid 211 | . 2 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
18 | 1, 17 | bitrid 283 | 1 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 class class class wbr 5106 ◡ccnv 5633 Rel wrel 5639 Rels crels 36682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-rels 36993 |
This theorem is referenced by: elrelscnveq 37000 |
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