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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsymrels2 | Structured version Visualization version GIF version |
Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
Ref | Expression |
---|---|
elsymrels2 | ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsymrels2 35775 | . 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | |
2 | cnveq 5739 | . . 3 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
3 | id 22 | . . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
4 | 2, 3 | sseq12d 4000 | . 2 ⊢ (𝑟 = 𝑅 → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑅 ⊆ 𝑅)) |
5 | 1, 4 | rabeqel 35510 | 1 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ◡ccnv 5549 Rels crels 35449 SymRels csymrels 35458 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-rels 35719 df-ssr 35732 df-syms 35772 df-symrels 35773 |
This theorem is referenced by: elsymrelsrel 35787 |
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