Theorem refrelsredund2 38609

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38489) 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 38608	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2		df-eqvrels 38560	. . . 4 ⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3		inss1 4190	. . . 4 ⊢ (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
4	2, 3	eqsstri 3984	. . 3 ⊢ EqvRels ⊆ ( RefRels ∩ SymRels )
5	4	redundss3 38604	. 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 3396   ∩ cin 3904   ⊆ wss 3905   I cid 5517  dom cdm 5623   ↾ cres 5625   Rels crels 38156   RefRels crefrels 38159   SymRels csymrels 38165   TrRels ctrrels 38168   EqvRels ceqvrels 38170   Redund wredund 38175

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-rels 38461  df-ssr 38474  df-refs 38486  df-refrels 38487  df-syms 38518  df-symrels 38519  df-eqvrels 38560  df-redund 38600

This theorem is referenced by:  refrelsredund3  38610