Theorem detxrnidres 39648

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 39606	. 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
2	1	detlem 39634	1 ⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)))

This proof depends on syntax axioms:   ↔ wb 209   I cid 5549   ↾ cres 5657   ⋉ cxrn 38922   ≀ ccoss 38931   EqvRel weqvrel 38948   Disj wdisjALTV 38967

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7986  df-2nd 7987  df-ec 8698  df-xrn 39128  df-coss 39249  df-refrel 39340  df-cnvrefrel 39355  df-symrel 39372  df-trrel 39406  df-eqvrel 39417  df-funALTV 39515  df-disjALTV 39538

This theorem is used by: (None)