Theorem refrelsredund2 39617

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 39493) 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 39616	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2		df-eqvrels 39568	. . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3		inss1 4182	. . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
4	2, 3	eqsstri 3977	. . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels )
5	4	redundss3 39612	. 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 ⟩

This proof depends on syntax axioms:  {crab 3413   ∩ cin 3898   ⊆ wss 3899   I cid 5545  dom cdm 5651   ↾ cres 5653   Rels crels 39085   RefRels crefrels 39088   SymRels csymrels 39094   TrRels ctrrels 39097   EqvRels ceqvrels 39099   Redund wredund 39104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-rels 39340  df-ssr 39478  df-refs 39490  df-refrels 39491  df-syms 39522  df-symrels 39523  df-eqvrels 39568  df-redund 39608

This theorem is used by:  refrelsredund3  39618