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Theorem refrelredund2 38160
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38039) 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 38159 . 2 redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
2 df-eqvrel 38109 . . . 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 216 . . 3 ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅))
54redundpim3 38154 . 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 394  w3a 1084  wss 3941   I cid 5570  dom cdm 5673  cres 5675  Rel wrel 5678   RefRel wrefrel 37707   SymRel wsymrel 37713   TrRel wtrrel 37716   EqvRel weqvrel 37718   redund wredundp 37723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-refrel 38036  df-symrel 38068  df-eqvrel 38109  df-redundp 38149
This theorem is referenced by:  refrelredund3  38161
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