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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 35913) 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 36027 | . 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , ( RefRels ∩ SymRels )〉 | |
2 | df-eqvrels 35979 | . . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels ) | |
3 | inss1 4155 | . . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels ) | |
4 | 2, 3 | eqsstri 3949 | . . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels ) |
5 | 4 | redundss3 36023 | . 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 3110 ∩ cin 3880 ⊆ wss 3881 I cid 5424 dom cdm 5519 ↾ cres 5521 Rels crels 35615 RefRels crefrels 35618 SymRels csymrels 35624 TrRels ctrrels 35627 EqvRels ceqvrels 35629 Redund wredund 35634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-rels 35885 df-ssr 35898 df-refs 35910 df-refrels 35911 df-syms 35938 df-symrels 35939 df-eqvrels 35979 df-redund 36019 |
This theorem is referenced by: refrelsredund3 36029 |
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