Theorem petlemi 38771

Description: If you can prove disjointness (e.g. disjALTV0 38712, disjALTVid 38713, disjALTVidres 38714, disjALTVxrnidres 38716, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 38671), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. (Contributed by Peter Mazsa, 18-Sep-2021.)

Hypothesis

Ref	Expression
petlemi.1	⊢ Disj 𝑅

Assertion

Ref	Expression
petlemi	⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Proof of Theorem petlemi

Step	Hyp	Ref	Expression
1		petlemi.1	. . 3 ⊢ Disj 𝑅
2	1	a1i 11	. 2 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → Disj 𝑅)
3	2	petlem 38770	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  dom cdm 5700   / cqs 8764   ≀ ccoss 38137   EqvRel weqvrel 38154   Disj wdisjALTV 38171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767  df-qs 8771  df-coss 38369  df-refrel 38470  df-cnvrefrel 38485  df-symrel 38502  df-trrel 38532  df-eqvrel 38543  df-disjALTV 38663

This theorem is referenced by:  pet02  38772  petid2  38774  petidres2  38776  petinidres2  38778  petxrnidres2  38780