Theorem petlemi 39196

Description: If you can prove disjointness (e.g. disjALTV0 39134, disjALTVid 39135, disjALTVidres 39136, disjALTVxrnidres 39138, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 39078), 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 39195	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  dom cdm 5634   / cqs 8646   ≀ ccoss 38463   EqvRel weqvrel 38480   Disj wdisjALTV 38499

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8649  df-qs 8653  df-coss 38781  df-refrel 38872  df-cnvrefrel 38887  df-symrel 38904  df-trrel 38938  df-eqvrel 38949  df-disjALTV 39070

This theorem is referenced by:  pet02  39197  petid2  39199  petidres2  39201  petinidres2  39203  petxrnidres2  39205