Theorem detxrnidres 37662

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

Assertion

Ref	Expression
detxrnidres	⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)))

Proof of Theorem detxrnidres

Step	Hyp	Ref	Expression
1		disjALTVxrnidres 37623	. 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
2	1	detlem 37648	1 ⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)))

Syntax hints:   ↔ wb 205   I cid 5573   ↾ cres 5678   ⋉ cxrn 37037   ≀ ccoss 37038   EqvRel weqvrel 37055   Disj wdisjALTV 37072

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7974  df-2nd 7975  df-ec 8704  df-xrn 37236  df-coss 37276  df-refrel 37377  df-cnvrefrel 37392  df-symrel 37409  df-trrel 37439  df-eqvrel 37450  df-funALTV 37547  df-disjALTV 37570

This theorem is referenced by: (None)