Theorem detinidres 38788

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 38749	. 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
2	1	detlem 38775	1 ⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))

Syntax hints:   ↔ wb 206   ∩ cin 3913   I cid 5532   ↾ cres 5640   ≀ ccoss 38169   EqvRel weqvrel 38186   Disj wdisjALTV 38203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-coss 38402  df-refrel 38503  df-cnvrefrel 38518  df-symrel 38535  df-trrel 38565  df-eqvrel 38576  df-funALTV 38674  df-disjALTV 38697

This theorem is referenced by: (None)