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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 37004) 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 37123 | . 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩ | |
2 | df-eqvrels 37075 | . . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels ) | |
3 | inss1 4193 | . . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels ) | |
4 | 2, 3 | eqsstri 3983 | . . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels ) |
5 | 4 | redundss3 37119 | . 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 3410 ∩ cin 3914 ⊆ wss 3915 I cid 5535 dom cdm 5638 ↾ cres 5640 Rels crels 36665 RefRels crefrels 36668 SymRels csymrels 36674 TrRels ctrrels 36677 EqvRels ceqvrels 36679 Redund wredund 36684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-rels 36976 df-ssr 36989 df-refs 37001 df-refrels 37002 df-syms 37033 df-symrels 37034 df-eqvrels 37075 df-redund 37115 |
This theorem is referenced by: refrelsredund3 37125 |
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