Theorem erim2 35442

Description: Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 35549 in a more convenient form , see also erim 35443). (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
erim2	⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Proof of Theorem erim2

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

Step	Hyp	Ref	Expression
1		relcoels 35200	. . . 4 ⊢ Rel ∼ 𝐴
2	1	a1i 11	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel ∼ 𝐴)
3		eqvrelrel 35363	. . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
4	3	ad2antrl 724	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → Rel 𝑅)
5		brcoels 35211	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)))
6	5	el2v 3444	. . . 4 ⊢ (𝑥 ∼ 𝐴𝑦 ↔ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢))
7		simpll 763	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → EqvRel 𝑅)
8		simprl 767	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ 𝐴)
9		simplr 765	. . . . . . . . . . . . 13 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (dom 𝑅 / 𝑅) = 𝐴)
10	8, 9	eleqtrrd 2886	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 ∈ (dom 𝑅 / 𝑅))
11		simprr 769	. . . . . . . . . . . 12 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑥 ∈ 𝑢)
12		eqvrelqsel 35382	. . . . . . . . . . . 12 ⊢ (( EqvRel 𝑅 ∧ 𝑢 ∈ (dom 𝑅 / 𝑅) ∧ 𝑥 ∈ 𝑢) → 𝑢 = [𝑥]𝑅)
13	7, 10, 11, 12	syl3anc 1364	. . . . . . . . . . 11 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → 𝑢 = [𝑥]𝑅)
14	13	eleq2d 2868	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ [𝑥]𝑅))
15		elecALTV 35059	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦))
16	15	el2v 3444	. . . . . . . . . 10 ⊢ (𝑦 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝑦)
17	14, 16	syl6bb 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 579	. . . . . . 7 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑢 ∈ 𝐴) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
20	19	rexbidva 3259	. . . . . 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 763	. . . . . . . . . . 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 35378	. . . . . . . . . 10 ⊢ ((( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
25	24	adantll 710	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
26		eqvrelim 35367	. . . . . . . . . . . . . 14 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
27	26	ad2antrl 724	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → dom 𝑅 = ran 𝑅)
28	27	eleq2d 2868	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ ran 𝑅))
29		dmqseqim2 35422	. . . . . . . . . . . . . 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 4749	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
34	32, 33	syl6bb 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 563	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥𝑅𝑦 ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦)))
39		r19.41v 3308	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ∧ 𝑥𝑅𝑦))
40	38, 39	syl6bbr 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	syl5bb 284	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) → (𝑥 ∼ 𝐴𝑦 ↔ 𝑥𝑅𝑦))
43	2, 4, 42	eqbrrdv 5552	. 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 1522   ∈ wcel 2081  ∃wrex 3106  Vcvv 3437  ∪ cuni 4745   class class class wbr 4962  dom cdm 5443  ran crn 5444  Rel wrel 5448  [cec 8137   / cqs 8138   ∼ ccoels 34986   EqvRel weqvrel 35002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-id 5348  df-eprel 5353  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ec 8141  df-qs 8145  df-coss 35190  df-coels 35191  df-refrel 35283  df-symrel 35311  df-trrel 35341  df-eqvrel 35351

This theorem is referenced by:  erim  35443