Theorem petlemi 39257

Description: If you can prove disjointness (e.g. disjALTV0 39195, disjALTVid 39196, disjALTVidres 39197, disjALTVxrnidres 39199, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 39139), 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 39256	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  dom cdm 5626   / cqs 8637   ≀ ccoss 38524   EqvRel weqvrel 38541   Disj wdisjALTV 38560

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 5232  ax-pr 5372

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 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ec 8640  df-qs 8644  df-coss 38842  df-refrel 38933  df-cnvrefrel 38948  df-symrel 38965  df-trrel 38999  df-eqvrel 39010  df-disjALTV 39131

This theorem is referenced by:  pet02  39258  petid2  39260  petidres2  39262  petinidres2  39264  petxrnidres2  39266