Theorem refrelredund3 38660

Description: The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 38539) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)

Assertion

Ref	Expression
refrelredund3	⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Distinct variable group:   𝑥,𝑅

Proof of Theorem refrelredund3

Step	Hyp	Ref	Expression
1		refrelredund2 38659	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
2		idrefALT 6105	. . . 4 ⊢ (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
3	2	anbi1i 624	. . 3 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅))
4	3	redundpbi1 38654	. 2 ⊢ ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) ↔ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅))
5	1, 4	mpbi 230	1 ⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Syntax hints:   ∧ wa 395  ∀wral 3052   ⊆ wss 3931   class class class wbr 5124   I cid 5552  dom cdm 5659   ↾ cres 5661  Rel wrel 5664   RefRel wrefrel 38210   EqvRel weqvrel 38221   redund wredundp 38226

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-11 2158  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-refrel 38535  df-symrel 38567  df-eqvrel 38608  df-redundp 38648

This theorem is referenced by: (None)