Theorem detidres 38832

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 38793	. 2 ⊢ Disj ( I ↾ 𝐴)
2	1	detlem 38820	1 ⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))

Syntax hints:   ↔ wb 206   I cid 5510   ↾ cres 5618   ≀ ccoss 38214   EqvRel weqvrel 38231   Disj wdisjALTV 38248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-coss 38447  df-refrel 38548  df-cnvrefrel 38563  df-symrel 38580  df-trrel 38610  df-eqvrel 38621  df-funALTV 38719  df-disjALTV 38742

This theorem is referenced by: (None)