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Theorem refrelredund2 35936
 Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35820) 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 35935 . 2 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
2 df-eqvrel 35885 . . . 4 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
3 3simpa 1145 . . . 4 (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅))
42, 3sylbi 220 . . 3 ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅))
54redundpim3 35930 . 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 399   ∧ w3a 1084   ⊆ wss 3918   I cid 5442  dom cdm 5538   ↾ cres 5540  Rel wrel 5543   RefRel wrefrel 35524   SymRel wsymrel 35530   TrRel wtrrel 35533   EqvRel weqvrel 35535   redund wredundp 35540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5050  df-opab 5112  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-dm 5548  df-rn 5549  df-res 5550  df-refrel 35817  df-symrel 35845  df-eqvrel 35885  df-redundp 35925 This theorem is referenced by:  refrelredund3  35937
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