Theorem refrelsredund2 39180

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 39056) 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 39179	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2		df-eqvrels 39131	. . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3		inss1 4188	. . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
4	2, 3	eqsstri 3982	. . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels )
5	4	redundss3 39175	. 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 3413   ∩ cin 3903   ⊆ wss 3904   I cid 5539  dom cdm 5645   ↾ cres 5647   Rels crels 38648   RefRels crefrels 38651   SymRels csymrels 38657   TrRels ctrrels 38660   EqvRels ceqvrels 38662   Redund wredund 38667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-rels 38903  df-ssr 39041  df-refs 39053  df-refrels 39054  df-syms 39085  df-symrels 39086  df-eqvrels 39131  df-redund 39171

This theorem is referenced by:  refrelsredund3  39181