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Mirrors > Home > MPE Home > Th. List > Mathboxes > refsymrels3 | Structured version Visualization version GIF version |
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 35826) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 35756, cf. the comment of dfrefrel3 35758. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
refsymrels3 | ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refsymrels2 35803 | . 2 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} | |
2 | idrefALT 5975 | . . 3 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥) | |
3 | cnvsym 5976 | . . 3 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)) | |
4 | 2, 3 | anbi12i 628 | . 2 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))) |
5 | 1, 4 | rabbieq 35514 | 1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∀wral 3140 {crab 3144 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 I cid 5461 ◡ccnv 5556 dom cdm 5557 ↾ cres 5559 Rels crels 35457 RefRels crefrels 35460 SymRels csymrels 35466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-rels 35727 df-ssr 35740 df-refs 35752 df-refrels 35753 df-syms 35780 df-symrels 35781 |
This theorem is referenced by: (None) |
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