Theorem petlemi 39942

Description: If you can prove disjointness (e.g. disjALTV0 39880, disjALTVid 39881, disjALTVidres 39882, disjALTVxrnidres 39884, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 39824), 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 39941	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  dom cdm 5650   / cqs 8720   ≀ ccoss 39209   EqvRel weqvrel 39226   Disj wdisjALTV 39245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8723  df-qs 8727  df-coss 39527  df-refrel 39618  df-cnvrefrel 39633  df-symrel 39650  df-trrel 39684  df-eqvrel 39695  df-disjALTV 39816

This theorem is used by:  pet02  39943  petid2  39945  petidres2  39947  petinidres2  39949  petxrnidres2  39951