Theorem petlemi 39511

Description: If you can prove disjointness (e.g. disjALTV0 39449, disjALTVid 39450, disjALTVidres 39451, disjALTVxrnidres 39453, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 39393), 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 39510	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568  dom cdm 5661   / cqs 8692   ≀ ccoss 38778   EqvRel weqvrel 38795   Disj wdisjALTV 38814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8695  df-qs 8699  df-coss 39096  df-refrel 39187  df-cnvrefrel 39202  df-symrel 39219  df-trrel 39253  df-eqvrel 39264  df-disjALTV 39385

This theorem is referenced by:  pet02  39512  petid2  39514  petidres2  39516  petinidres2  39518  petxrnidres2  39520