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Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelsredund3 | Structured version Visualization version GIF version |
Description: The naive version of the class of reflexive relations {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥} is redundant with respect to the class of reflexive relations (see dfrefrels3 35788) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.) |
Ref | Expression |
---|---|
refrelsredund3 | ⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund 〈 RefRels , EqvRels 〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refrelsredund2 35902 | . 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉 | |
2 | idrefALT 5966 | . . . 4 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥) | |
3 | 2 | rabbii 3470 | . . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} |
4 | 3 | redundeq1 35898 | . 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉 ↔ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund 〈 RefRels , EqvRels 〉) |
5 | 1, 4 | mpbi 232 | 1 ⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund 〈 RefRels , EqvRels 〉 |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3137 {crab 3141 ⊆ wss 3929 class class class wbr 5059 I cid 5452 dom cdm 5548 ↾ cres 5550 Rels crels 35489 RefRels crefrels 35492 EqvRels ceqvrels 35503 Redund wredund 35508 |
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 35759 df-ssr 35772 df-refs 35784 df-refrels 35785 df-syms 35812 df-symrels 35813 df-eqvrels 35853 df-redund 35893 |
This theorem is referenced by: (None) |
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