Theorem petlem 38777

Description: If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 38798), or converse function (cf. dfdisjALTV 38678), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 38815. (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 38772	. 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 484	. . 3 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)
4		disjdmqseq 38770	. . . 4 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
5	4	pm5.32i 574	. . 3 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
6	2, 3, 5	sylanbrc 583	. 2 ⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
7	1, 6	impbii 209	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  dom cdm 5631   / cqs 8647   ≀ ccoss 38142   EqvRel weqvrel 38159   Disj wdisjALTV 38176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rmo 3351  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654  df-coss 38375  df-refrel 38476  df-cnvrefrel 38491  df-symrel 38508  df-trrel 38538  df-eqvrel 38549  df-disjALTV 38670

This theorem is referenced by:  petlemi  38778  mpet3  38801  cpet2  38802  petincnvepres2  38813  pet2  38815