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Theorem refrelredund2 37970
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 37849) 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
StepHypRef Expression
1 refrelredund4 37969 . 2 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
2 df-eqvrel 37919 . . . 4 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
3 3simpa 1147 . . . 4 (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅))
42, 3sylbi 216 . . 3 ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅))
54redundpim3 37964 . 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))
61, 5ax-mp 5 1 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086  wss 3948   I cid 5573  dom cdm 5676  cres 5678  Rel wrel 5681   RefRel wrefrel 37513   SymRel wsymrel 37519   TrRel wtrrel 37522   EqvRel weqvrel 37524   redund wredundp 37529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-refrel 37846  df-symrel 37878  df-eqvrel 37919  df-redundp 37959
This theorem is referenced by:  refrelredund3  37971
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