Theorem petlem 39411

Description: If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 39443), or converse function (cf. dfdisjALTV 39294), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 39460. (Contributed by Peter Mazsa, 18-Sep-2021.)

Hypothesis

Ref	Expression
petlem.1	⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → Disj 𝑅)

Assertion

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

Proof of Theorem petlem

Step	Hyp	Ref	Expression
1		partim2 39406	. 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
2		petlem.1	. . 3 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → Disj 𝑅)
3		simpr 488	. . 3 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)
4		disjdmqseq 39404	. . . 4 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
5	4	pm5.32i 582	. . 3 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
6	2, 3, 5	sylanbrc 592	. 2 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
7	1, 6	impbii 211	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560  dom cdm 5647   / cqs 8677   ≀ ccoss 38679   EqvRel weqvrel 38696   Disj wdisjALTV 38715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-qs 8684  df-coss 38997  df-refrel 39088  df-cnvrefrel 39103  df-symrel 39120  df-trrel 39154  df-eqvrel 39165  df-disjALTV 39286

This theorem is referenced by:  petlemi  39412  mpet3  39446  cpet2  39447  petincnvepres2  39458  pet2  39460