Theorem refrelredund3 39845

Description: The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 39720) 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 39844	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
2		idrefALT 6119	. . . 4 ⊢ (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
3	2	anbi1i 636	. . 3 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅))
4	3	redundpbi1 39839	. 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 233	1 ⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

This proof depends on syntax axioms:   ∧ wa 401  ∀wral 3080   ⊆ wss 3900   class class class wbr 5106   I cid 5549  dom cdm 5655   ↾ cres 5657  Rel wrel 5660   RefRel wrefrel 39313   EqvRel weqvrel 39324   redund wredundp 39329

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 5252  ax-pr 5395

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-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-refrel 39716  df-symrel 39748  df-eqvrel 39793  df-redundp 39833

This theorem is used by: (None)