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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 3998 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)) | |
| 2 | cnvsym 6131 | . . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
| 3 | 2 | biimpi 216 | . . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | 
| 4 | 3 | a1d 25 | . . . . 5 ⊢ (◡𝑅 ⊆ 𝑅 → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | 
| 5 | 4 | adantl 481 | . . . 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 38490 | . . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅) | |
| 8 | dfrel2 6208 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
| 9 | 7, 8 | sylib 218 | . . . . 5 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅) | 
| 10 | cnvss 5882 | . . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅) | |
| 11 | sseq1 4008 | . . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅)) | |
| 12 | 10, 11 | syl5ibcom 245 | . . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅)) | 
| 13 | 2, 12 | sylbir 235 | . . . . 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 228 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅) | 
| 16 | 14, 15 | jca2 513 | . . 3 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅))) | 
| 17 | 6, 16 | impbid 212 | . 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 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 ◡ccnv 5683 Rel wrel 5689 Rels crels 38185 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-rels 38487 | 
| This theorem is referenced by: elrelscnveq 38494 | 
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