Theorem detidres 38730

Description: The cosets by the restricted identity relation are in equivalence relation if and only if the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
detidres	⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))

Proof of Theorem detidres

Step	Hyp	Ref	Expression
1		disjALTVidres 38691	. 2 ⊢ Disj ( I ↾ 𝐴)
2	1	detlem 38718	1 ⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))

Syntax hints:   ↔ wb 206   I cid 5557   ↾ cres 5667   ≀ ccoss 38116   EqvRel weqvrel 38133   Disj wdisjALTV 38150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-coss 38346  df-refrel 38447  df-cnvrefrel 38462  df-symrel 38479  df-trrel 38509  df-eqvrel 38520  df-funALTV 38617  df-disjALTV 38640

This theorem is referenced by: (None)