Theorem refrelsredund3 39178

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 39054) 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 39177	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
2		idrefALT 6096	. . . 4 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3	2	rabbii 3418	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥}
4	3	redundeq1 39173	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩ ↔ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩)
5	1, 4	mpbi 232	1 ⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩

Syntax hints:  ∀wral 3075  {crab 3413   ⊆ wss 3902   class class class wbr 5097   I cid 5537  dom cdm 5643   ↾ cres 5645   Rels crels 38645   RefRels crefrels 38648   EqvRels ceqvrels 38659   Redund wredund 38664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-rels 38900  df-ssr 39038  df-refs 39050  df-refrels 39051  df-syms 39082  df-symrels 39083  df-eqvrels 39128  df-redund 39168

This theorem is referenced by: (None)