Theorem partim2 39248

Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 39249. Lemma for petlem 39253. (Contributed by Peter Mazsa, 17-Sep-2021.)

Assertion

Ref	Expression
partim2	⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Proof of Theorem partim2

Step	Hyp	Ref	Expression
1		disjim 39222	. . 3 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
2	1	adantr 480	. 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → EqvRel ≀ 𝑅)
3		disjdmqseq 39246	. . 3 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
4	3	biimpa 476	. 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)
5	2, 4	jca 511	1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  dom cdm 5625   / cqs 8636   ≀ ccoss 38521   EqvRel weqvrel 38538   Disj wdisjALTV 38557

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 5371

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 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643  df-coss 38839  df-refrel 38930  df-cnvrefrel 38945  df-symrel 38962  df-trrel 38996  df-eqvrel 39007  df-disjALTV 39128

This theorem is referenced by:  partim  39249  petlem  39253