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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 | eqcom 2825 | . 2 ⊢ (𝑅 = ◡𝑅 ↔ ◡𝑅 = 𝑅) | |
2 | relcnveq3 35459 | . . 3 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | |
3 | cnvsym 5967 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) | |
4 | 2, 3 | syl6bbr 290 | . 2 ⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ◡𝑅 ⊆ 𝑅)) |
5 | 1, 4 | syl5rbbr 287 | 1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ⊆ wss 3933 class class class wbr 5057 ◡ccnv 5547 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 |
This theorem is referenced by: relcnveq4 35462 cnvcosseq 35562 dfsymrel4 35667 |
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