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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfsymrels3 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
Ref | Expression |
---|---|
dfsymrels3 | ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrels2 38526 | . 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | |
2 | cnvsym 6134 | . 2 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)) | |
3 | 1, 2 | rabbieq 3441 | 1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 = wceq 1536 {crab 3432 ⊆ wss 3962 class class class wbr 5147 ◡ccnv 5687 Rels crels 38163 SymRels csymrels 38172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-rels 38466 df-ssr 38479 df-syms 38523 df-symrels 38524 |
This theorem is referenced by: elsymrels3 38535 |
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