Theorem erimeq2 39139

Description: Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 39383 in a more convenient form , see also erimeq 39140). (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 38890	. . . 4 ⊢ Rel ∼ 𝐴
2	1	a1i 11	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel ∼ 𝐴)
3		eqvrelrel 39057	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4	3	ad2antrl 734	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel 𝑅)
5		brcoels 38901	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)))
6	5	el2v 3438	. . . 4 ⊢ (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢))
7		simpll 772	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → EqvRel 𝑅)
8		simprl 776	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ 𝐴)
9		simplr 774	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (dom 𝑅 / 𝑅) = 𝐴)
10	8, 9	eleqtrrd 2842	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ (dom 𝑅 / 𝑅))
11		simprr 778	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑥 ∈ 𝑢)
12		eqvrelqsel 39076	. . . . . . . . . . . 12 ⊢ (( EqvRel 𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑥 ∈ 𝑢) → 𝑢 = [𝑥]𝑅)
13	7, 10, 11, 12	syl3anc 1379	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 = [𝑥]𝑅)
14	13	eleq2d 2825	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ [𝑥]𝑅))
15		elecALTV 38647	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦))
16	15	el2v 3438	. . . . . . . . . 10 ⊢ (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦)
17	14, 16	bitrdi 288	. . . . . . . . 9 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (𝑦 ∈ 𝑢 ↔ 𝑥𝑅𝑦))
18	17	anassrs 468	. . . . . . . 8 ⊢ (((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑢 ∈ 𝐴) ∧ 𝑥 ∈ 𝑢) → (𝑦 ∈ 𝑢 ↔ 𝑥𝑅𝑦))
19	18	pm5.32da 584	. . . . . . 7 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑢 ∈ 𝐴) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
20	19	rexbidva 3161	. . . . . 6 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
21	20	adantl 482	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
22		simpll 772	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → EqvRel 𝑅)
23		simpr 485	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
24	22, 23	eqvrelcl 39072	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
25	24	adantll 720	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
26		eqvrelim 39061	. . . . . . . . . . . . . 14 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
27	26	ad2antrl 734	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → dom 𝑅 = ran 𝑅)
28	27	eleq2d 2825	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ran 𝑅))
29		dmqseqim2 39118	. . . . . . . . . . . . . 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 419	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ ran 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))
32	28, 31	bitrd 280	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ∪ 𝐴))
33		eluni2 4843	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
34	32, 33	bitrdi 288	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
35	34	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥 ∈ dom 𝑅 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
36	25, 35	mpbid 233	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
37	36	ex 413	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 → ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
38	37	pm4.71rd 567	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
39		r19.41v 3169	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦))
40	38, 39	bitr4di 290	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
41	21, 40	bitr4d 283	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ 𝑥𝑅𝑦))
42	6, 41	bitrid 284	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∼ 𝐴𝑦 ↔ 𝑥𝑅𝑦))
43	2, 4, 42	eqbrrdv 5737	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → ∼ 𝐴 = 𝑅)
44	43	ex 413	1 ⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431  ∪ cuni 4839   class class class wbr 5073  dom cdm 5619  ran crn 5620  Rel wrel 5624  [cec 8632   / cqs 8633   ∼ ccoels 38560   EqvRel weqvrel 38576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2711  ax-sep 5219  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8636  df-qs 8640  df-coss 38877  df-coels 38878  df-refrel 38968  df-symrel 39000  df-trrel 39034  df-eqvrel 39045

This theorem is referenced by:  erimeq  39140