Theorem detidres 39233

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 39191	. 2 ⊢ Disj ( I ↾ 𝐴)
2	1	detlem 39221	1 ⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))

Syntax hints:   ↔ wb 206   I cid 5518   ↾ cres 5626   ≀ ccoss 38518   EqvRel weqvrel 38535   Disj wdisjALTV 38554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-coss 38836  df-refrel 38927  df-cnvrefrel 38942  df-symrel 38959  df-trrel 38993  df-eqvrel 39004  df-funALTV 39102  df-disjALTV 39125

This theorem is referenced by: (None)