Theorem erimeq2 38015

Description: Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 38219 in a more convenient form , see also erimeq 38016). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.)

Assertion

Ref	Expression
erimeq2	⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Proof of Theorem erimeq2

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcoels 37761	. . . 4 ⊢ Rel ∼ 𝐴
2	1	a1i 11	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel ∼ 𝐴)
3		eqvrelrel 37934	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4	3	ad2antrl 725	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel 𝑅)
5		brcoels 37772	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)))
6	5	el2v 3481	. . . 4 ⊢ (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢))
7		simpll 764	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → EqvRel 𝑅)
8		simprl 768	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ 𝐴)
9		simplr 766	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (dom 𝑅 / 𝑅) = 𝐴)
10	8, 9	eleqtrrd 2835	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ (dom 𝑅 / 𝑅))
11		simprr 770	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑥 ∈ 𝑢)
12		eqvrelqsel 37953	. . . . . . . . . . . 12 ⊢ (( EqvRel 𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑥 ∈ 𝑢) → 𝑢 = [𝑥]𝑅)
13	7, 10, 11, 12	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 = [𝑥]𝑅)
14	13	eleq2d 2818	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ [𝑥]𝑅))
15		elecALTV 37601	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦))
16	15	el2v 3481	. . . . . . . . . 10 ⊢ (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦)
17	14, 16	bitrdi 287	. . . . . . . . 9 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (𝑦 ∈ 𝑢 ↔ 𝑥𝑅𝑦))
18	17	anassrs 467	. . . . . . . 8 ⊢ (((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑢 ∈ 𝐴) ∧ 𝑥 ∈ 𝑢) → (𝑦 ∈ 𝑢 ↔ 𝑥𝑅𝑦))
19	18	pm5.32da 578	. . . . . . 7 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑢 ∈ 𝐴) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
20	19	rexbidva 3175	. . . . . 6 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
21	20	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
22		simpll 764	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → EqvRel 𝑅)
23		simpr 484	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
24	22, 23	eqvrelcl 37949	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
25	24	adantll 711	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
26		eqvrelim 37938	. . . . . . . . . . . . . 14 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
27	26	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → dom 𝑅 = ran 𝑅)
28	27	eleq2d 2818	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ran 𝑅))
29		dmqseqim2 37994	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝑥 ∈ ran 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))))
30	3, 29	syl5 34	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝑉 → ( EqvRel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝑥 ∈ ran 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))))
31	30	imp32 418	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ ran 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))
32	28, 31	bitrd 279	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))
33		eluni2 4912	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
34	32, 33	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
35	34	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥 ∈ dom 𝑅 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
36	25, 35	mpbid 231	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
37	36	ex 412	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 → ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
38	37	pm4.71rd 562	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
39		r19.41v 3187	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦))
40	38, 39	bitr4di 289	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
41	21, 40	bitr4d 282	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ 𝑥𝑅𝑦))
42	6, 41	bitrid 283	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∼ 𝐴𝑦 ↔ 𝑥𝑅𝑦))
43	2, 4, 42	eqbrrdv 5793	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → ∼ 𝐴 = 𝑅)
44	43	ex 412	1 ⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473  ∪ cuni 4908   class class class wbr 5148  dom cdm 5676  ran crn 5677  Rel wrel 5681  [cec 8707   / cqs 8708   ∼ ccoels 37511   EqvRel weqvrel 37527

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711  df-qs 8715  df-coss 37748  df-coels 37749  df-refrel 37849  df-symrel 37881  df-trrel 37911  df-eqvrel 37922

This theorem is referenced by:  erimeq  38016