Theorem refrelredund3 39746

Description: The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 39621) 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 39745	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
2		idrefALT 6106	. . . 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 39740	. 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 3076   ⊆ wss 3895   class class class wbr 5102   I cid 5544  dom cdm 5650   ↾ cres 5652  Rel wrel 5655   RefRel wrefrel 39214   EqvRel weqvrel 39225   redund wredundp 39230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-refrel 39617  df-symrel 39649  df-eqvrel 39694  df-redundp 39734

This theorem is used by: (None)