Theorem detinidres 38790

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

Assertion

Ref	Expression
detinidres	⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))

Proof of Theorem detinidres

Step	Hyp	Ref	Expression
1		disjALTVinidres 38751	. 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
2	1	detlem 38777	1 ⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))

Syntax hints:   ↔ wb 206   ∩ cin 3963   I cid 5583   ↾ cres 5692   ≀ ccoss 38174   EqvRel weqvrel 38191   Disj wdisjALTV 38208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  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 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-coss 38405  df-refrel 38506  df-cnvrefrel 38521  df-symrel 38538  df-trrel 38568  df-eqvrel 38579  df-funALTV 38676  df-disjALTV 38699

This theorem is referenced by: (None)