Theorem refrelsredund3 38753

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 38629) 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 38752	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
2		idrefALT 6066	. . . 4 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3	2	rabbii 3401	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥}
4	3	redundeq1 38748	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩ ↔ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩)
5	1, 4	mpbi 230	1 ⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩

Syntax hints:  ∀wral 3048  {crab 3396   ⊆ wss 3898   class class class wbr 5095   I cid 5515  dom cdm 5621   ↾ cres 5623   Rels crels 38247   RefRels crefrels 38250   EqvRels ceqvrels 38261   Redund wredund 38266

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  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 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-rels 38487  df-ssr 38613  df-refs 38625  df-refrels 38626  df-syms 38657  df-symrels 38658  df-eqvrels 38703  df-redund 38743

This theorem is referenced by: (None)