Theorem refrelsredund2 35900

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35785) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)

Assertion

Ref	Expression
refrelsredund2	⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩

Proof of Theorem refrelsredund2

Step	Hyp	Ref	Expression
1		refrelsredund4 35899	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2		df-eqvrels 35851	. . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3		inss1 4193	. . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
4	2, 3	eqsstri 3989	. . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels )
5	4	redundss3 35895	. 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 ⟩

Syntax hints:  {crab 3142   ∩ cin 3923   ⊆ wss 3924   I cid 5445  dom cdm 5541   ↾ cres 5543   Rels crels 35487   RefRels crefrels 35490   SymRels csymrels 35496   TrRels ctrrels 35499   EqvRels ceqvrels 35501   Redund wredund 35506

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 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-op 4560  df-br 5053  df-opab 5115  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-dm 5551  df-rn 5552  df-res 5553  df-rels 35757  df-ssr 35770  df-refs 35782  df-refrels 35783  df-syms 35810  df-symrels 35811  df-eqvrels 35851  df-redund 35891

This theorem is referenced by:  refrelsredund3  35901