Theorem prter3 36582

Description: For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prter3	⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆)

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)   𝑆(𝑥,𝑦,𝑢)

Proof of Theorem prter3

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		errel 8378	. . 3 ⊢ (𝑆 Er ∪ 𝐴 → Rel 𝑆)
2	1	adantr 484	. 2 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → Rel 𝑆)
3		prtlem18.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	relopabiv 5675	. . 3 ⊢ Rel ∼
5	3	prtlem13 36568	. . . . . 6 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑆 Er ∪ 𝐴)
7		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ 𝐴)
8		ne0i 4235	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑣 → 𝑣 ≠ ∅)
9	8	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ≠ ∅)
10		eldifsn 4686	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴 ∖ {∅}) ↔ (𝑣 ∈ 𝐴 ∧ 𝑣 ≠ ∅))
11	7, 9, 10	sylanbrc 586	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (𝐴 ∖ {∅}))
12		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))
13	11, 12	eleqtrrd 2834	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (∪ 𝐴 / 𝑆))
14		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑧 ∈ 𝑣)
15		qsel 8456	. . . . . . . . . . . . 13 ⊢ ((𝑆 Er ∪ 𝐴 ∧ 𝑣 ∈ (∪ 𝐴 / 𝑆) ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧]𝑆)
16	6, 13, 14, 15	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧]𝑆)
17	16	eleq2d 2816	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧]𝑆))
18		vex 3402	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
19		vex 3402	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	18, 19	elec 8413	. . . . . . . . . . 11 ⊢ (𝑤 ∈ [𝑧]𝑆 ↔ 𝑧𝑆𝑤)
21	17, 20	bitrdi 290	. . . . . . . . . 10 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑧𝑆𝑤))
22	21	anassrs 471	. . . . . . . . 9 ⊢ ((((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧𝑆𝑤))
23	22	pm5.32da 582	. . . . . . . 8 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
24	23	rexbidva 3205	. . . . . . 7 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
25		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑆 Er ∪ 𝐴)
26		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧𝑆𝑤)
27	25, 26	ercl 8380	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧 ∈ ∪ 𝐴)
28		eluni2 4809	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
29	27, 28	sylib 221	. . . . . . . . . 10 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
30	29	ex 416	. . . . . . . . 9 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 → ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
31	30	pm4.71rd 566	. . . . . . . 8 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
32		r19.41v 3250	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤))
33	31, 32	bitr4di 292	. . . . . . 7 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
34	24, 33	bitr4d 285	. . . . . 6 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧𝑆𝑤))
35	5, 34	syl5bb 286	. . . . 5 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
36	35	adantl 485	. . . 4 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
37	36	eqbrrdv2 36563	. . 3 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
38	4, 37	mpanl1 700	. 2 ⊢ ((Rel 𝑆 ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
39	2, 38	mpancom 688	1 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∃wrex 3052   ∖ cdif 3850  ∅c0 4223  {csn 4527  ∪ cuni 4805   class class class wbr 5039  {copab 5101  Rel wrel 5541   Er wer 8366  [cec 8367   / cqs 8368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-er 8369  df-ec 8371  df-qs 8375

This theorem is referenced by: (None)