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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsymrels5 | Structured version Visualization version GIF version |
Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
Ref | Expression |
---|---|
elsymrels5 | ⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrels5 37931 | . 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} | |
2 | breq 5143 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦)) | |
3 | breq 5143 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥)) | |
4 | 2, 3 | bibi12d 345 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
5 | 4 | 2albidv 1918 | . 2 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
6 | 1, 5 | rabeqel 37635 | 1 ⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 Rels crels 37558 SymRels csymrels 37567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-rels 37868 df-ssr 37881 df-syms 37925 df-symrels 37926 |
This theorem is referenced by: (None) |
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