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Theorem refrelsredund2 36000
Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35885) 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
StepHypRef Expression
1 refrelsredund4 35999 . 2 {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
2 df-eqvrels 35951 . . . 4 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
3 inss1 4190 . . . 4 (( RefRels ∩ SymRels ) ∩ TrRels ) ⊆ ( RefRels ∩ SymRels )
42, 3eqsstri 3987 . . 3 EqvRels ⊆ ( RefRels ∩ SymRels )
54redundss3 35995 . 2 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩ → {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩)
61, 5ax-mp 5 1 {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
Colors of variables: wff setvar class
Syntax hints:  {crab 3137  cin 3918  wss 3919   I cid 5447  dom cdm 5543  cres 5545   Rels crels 35587   RefRels crefrels 35590   SymRels csymrels 35596   TrRels ctrrels 35599   EqvRels ceqvrels 35601   Redund wredund 35606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-dm 5553  df-rn 5554  df-res 5555  df-rels 35857  df-ssr 35870  df-refs 35882  df-refrels 35883  df-syms 35910  df-symrels 35911  df-eqvrels 35951  df-redund 35991
This theorem is referenced by:  refrelsredund3  36001
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