Theorem detidres 39580

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 39538	. 2 ⊢ Disj ( I ↾ 𝐴)
2	1	detlem 39568	1 ⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   I cid 5557   ↾ cres 5665   ≀ ccoss 38865   EqvRel weqvrel 38882   Disj wdisjALTV 38901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-coss 39183  df-refrel 39274  df-cnvrefrel 39289  df-symrel 39306  df-trrel 39340  df-eqvrel 39351  df-funALTV 39449  df-disjALTV 39472

This theorem is used by: (None)