Theorem refrelsredund2 36872

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 36752) 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 36871	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2		df-eqvrels 36823	. . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3		inss1 4172	. . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
4	2, 3	eqsstri 3964	. . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels )
5	4	redundss3 36867	. 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 3403   ∩ cin 3895   ⊆ wss 3896   I cid 5505  dom cdm 5607   ↾ cres 5609   Rels crels 36412   RefRels crefrels 36415   SymRels csymrels 36421   TrRels ctrrels 36424   EqvRels ceqvrels 36426   Redund wredund 36431

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-br 5087  df-opab 5149  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-dm 5617  df-rn 5618  df-res 5619  df-rels 36724  df-ssr 36737  df-refs 36749  df-refrels 36750  df-syms 36781  df-symrels 36782  df-eqvrels 36823  df-redund 36863

This theorem is referenced by:  refrelsredund3  36873