Theorem refrelredund2 39746

Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 39621) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)

Assertion

Ref	Expression
refrelredund2	⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Proof of Theorem refrelredund2

Step	Hyp	Ref	Expression
1		refrelredund4 39745	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
2		df-eqvrel 39695	. . . 4 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
3		3simpa 1166	. . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅))
4	2, 3	sylbi 220	. . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅))
5	4	redundpim3 39740	. 2 ⊢ ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) → redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅))
6	1, 5	ax-mp 5	1 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   ⊆ wss 3895   I cid 5544  dom cdm 5650   ↾ cres 5652  Rel wrel 5655   RefRel wrefrel 39215   SymRel wsymrel 39221   TrRel wtrrel 39224   EqvRel weqvrel 39226   redund wredundp 39231

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-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 39618  df-symrel 39650  df-eqvrel 39695  df-redundp 39735

This theorem is used by:  refrelredund3  39747