Theorem refrelsredund3 39916

Description: The naive version of the class of reflexive relations {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥} is redundant with respect to the class of reflexive relations (see dfrefrels3 39792) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)

Assertion

Ref	Expression
refrelsredund3	⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩

Distinct variable group:   𝑥,𝑟

Proof of Theorem refrelsredund3

Step	Hyp	Ref	Expression
1		refrelsredund2 39915	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
2		idrefALT 6125	. . . 4 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3	2	rabbii 3421	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥}
4	3	redundeq1 39911	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩ ↔ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩)
5	1, 4	mpbi 233	1 ⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩

This proof depends on syntax axioms:  ∀wral 3080  {crab 3416   ⊆ wss 3900   class class class wbr 5108   I cid 5552  dom cdm 5658   ↾ cres 5660   Rels crels 39383   RefRels crefrels 39386   EqvRels ceqvrels 39397   Redund wredund 39402

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-rels 39638  df-ssr 39776  df-refs 39788  df-refrels 39789  df-syms 39820  df-symrels 39821  df-eqvrels 39866  df-redund 39906

This theorem is used by: (None)