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Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelsredund2 | Structured version Visualization version GIF version |
Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35786) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.) |
Ref | Expression |
---|---|
refrelsredund2 | ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refrelsredund4 35900 | . 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , ( RefRels ∩ SymRels )〉 | |
2 | df-eqvrels 35852 | . . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels ) | |
3 | inss1 4198 | . . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels ) | |
4 | 2, 3 | eqsstri 3994 | . . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels ) |
5 | 4 | redundss3 35896 | . 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , ( RefRels ∩ SymRels )〉 → {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉 |
Colors of variables: wff setvar class |
Syntax hints: {crab 3141 ∩ cin 3928 ⊆ wss 3929 I cid 5452 dom cdm 5548 ↾ cres 5550 Rels crels 35488 RefRels crefrels 35491 SymRels csymrels 35497 TrRels ctrrels 35500 EqvRels ceqvrels 35502 Redund wredund 35507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-rels 35758 df-ssr 35771 df-refs 35783 df-refrels 35784 df-syms 35811 df-symrels 35812 df-eqvrels 35852 df-redund 35892 |
This theorem is referenced by: refrelsredund3 35902 |
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