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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > relcnveq | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.) |
Ref | Expression |
---|---|
relcnveq | ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnveq3 38303 | . . 3 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | |
2 | cnvsym 6135 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
3 | 1, 2 | bitr4di 289 | . 2 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅)) |
4 | eqcom 2742 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅) | |
5 | 3, 4 | bitr3di 286 | 1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ⊆ wss 3963 class class class wbr 5148 ◡ccnv 5688 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: relcnveq4 38306 cnvcosseq 38419 dfsymrel4 38533 |
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