Theorem refrelredund2 38622

Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38501) 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 38621	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
2		df-eqvrel 38571	. . . 4 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
3		3simpa 1148	. . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅))
4	2, 3	sylbi 217	. . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅))
5	4	redundpim3 38616	. 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)

Syntax hints:   ∧ wa 395   ∧ w3a 1086   ⊆ wss 3916   I cid 5534  dom cdm 5640   ↾ cres 5642  Rel wrel 5645   RefRel wrefrel 38170   SymRel wsymrel 38176   TrRel wtrrel 38179   EqvRel weqvrel 38181   redund wredundp 38186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-refrel 38498  df-symrel 38530  df-eqvrel 38571  df-redundp 38611

This theorem is referenced by:  refrelredund3  38623