Theorem refrelredund3 38161

Description: The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 38040) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)

Assertion

Ref	Expression
refrelredund3	⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Distinct variable group:   𝑥,𝑅

Proof of Theorem refrelredund3

Step	Hyp	Ref	Expression
1		refrelredund2 38160	. 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
2		idrefALT 6113	. . . 4 ⊢ (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
3	2	anbi1i 622	. . 3 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅))
4	3	redundpbi1 38155	. 2 ⊢ ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) ↔ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅))
5	1, 4	mpbi 229	1 ⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Syntax hints:   ∧ wa 394  ∀wral 3051   ⊆ wss 3941   class class class wbr 5144   I cid 5570  dom cdm 5673   ↾ cres 5675  Rel wrel 5678   RefRel wrefrel 37707   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-11 2146  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: (None)