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Mirrors > Home > MPE Home > Th. List > Mathboxes > relcnveq3 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
relcnveq3 | ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3959 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)) | |
2 | cnvsym 6066 | . . . . . . 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 → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → (Rel 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
6 | 5 | com12 32 | . . 3 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
7 | dfrel2 6141 | . . . . 5 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
8 | cnvss 5828 | . . . . . . . 8 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅) | |
9 | sseq1 3969 | . . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅)) | |
10 | 8, 9 | syl5ibcom 244 | . . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅)) |
11 | 2, 10 | sylbir 234 | . . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅)) |
12 | 11 | com12 32 | . . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅)) |
13 | 7, 12 | sylbi 216 | . . . 4 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅)) |
14 | 2 | biimpri 227 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅) |
15 | 13, 14 | jca2 514 | . . 3 ⊢ (Rel 𝑅 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅))) |
16 | 6, 15 | impbid 211 | . 2 ⊢ (Rel 𝑅 → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
17 | 1, 16 | bitrid 282 | 1 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ⊆ wss 3910 class class class wbr 5105 ◡ccnv 5632 Rel wrel 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 |
This theorem is referenced by: relcnveq 36774 |
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