Theorem detxrnidres 38901

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 38862	. 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
2	1	detlem 38887	1 ⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)))

Syntax hints:   ↔ wb 206   I cid 5513   ↾ cres 5621   ⋉ cxrn 38220   ≀ ccoss 38228   EqvRel weqvrel 38245   Disj wdisjALTV 38262

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 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-1st 7927  df-2nd 7928  df-ec 8630  df-xrn 38410  df-coss 38519  df-refrel 38610  df-cnvrefrel 38625  df-symrel 38642  df-trrel 38676  df-eqvrel 38687  df-funALTV 38786  df-disjALTV 38809

This theorem is referenced by: (None)