Theorem petlem 39071

Description: If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 39092), or converse function (cf. dfdisjALTV 38972), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 39109. (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 39066	. 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 39064	. . . 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 1541  dom cdm 5624   / cqs 8634   ≀ ccoss 38383   EqvRel weqvrel 38400   Disj wdisjALTV 38417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641  df-coss 38674  df-refrel 38765  df-cnvrefrel 38780  df-symrel 38797  df-trrel 38831  df-eqvrel 38842  df-disjALTV 38964

This theorem is referenced by:  petlemi  39072  mpet3  39095  cpet2  39096  petincnvepres2  39107  pet2  39109