Theorem prter3 36823

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 8465	. . 3 ⊢ (𝑆 Er ∪ 𝐴 → Rel 𝑆)
2	1	adantr 480	. 2 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → Rel 𝑆)
3		prtlem18.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	relopabiv 5719	. . 3 ⊢ Rel ∼
5	3	prtlem13 36809	. . . . . 6 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		simpll 763	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑆 Er ∪ 𝐴)
7		simprl 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ 𝐴)
8		ne0i 4265	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑣 → 𝑣 ≠ ∅)
9	8	ad2antll 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ≠ ∅)
10		eldifsn 4717	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴 ∖ {∅}) ↔ (𝑣 ∈ 𝐴 ∧ 𝑣 ≠ ∅))
11	7, 9, 10	sylanbrc 582	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (𝐴 ∖ {∅}))
12		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))
13	11, 12	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (∪ 𝐴 / 𝑆))
14		simprr 769	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑧 ∈ 𝑣)
15		qsel 8543	. . . . . . . . . . . . 13 ⊢ ((𝑆 Er ∪ 𝐴 ∧ 𝑣 ∈ (∪ 𝐴 / 𝑆) ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧]𝑆)
16	6, 13, 14, 15	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧]𝑆)
17	16	eleq2d 2824	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧]𝑆))
18		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
19		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	18, 19	elec 8500	. . . . . . . . . . 11 ⊢ (𝑤 ∈ [𝑧]𝑆 ↔ 𝑧𝑆𝑤)
21	17, 20	bitrdi 286	. . . . . . . . . 10 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑧𝑆𝑤))
22	21	anassrs 467	. . . . . . . . 9 ⊢ ((((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧𝑆𝑤))
23	22	pm5.32da 578	. . . . . . . 8 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
24	23	rexbidva 3224	. . . . . . 7 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
25		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑆 Er ∪ 𝐴)
26		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧𝑆𝑤)
27	25, 26	ercl 8467	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧 ∈ ∪ 𝐴)
28		eluni2 4840	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
29	27, 28	sylib 217	. . . . . . . . . 10 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
30	29	ex 412	. . . . . . . . 9 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 → ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
31	30	pm4.71rd 562	. . . . . . . 8 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
32		r19.41v 3273	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤))
33	31, 32	bitr4di 288	. . . . . . 7 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
34	24, 33	bitr4d 281	. . . . . 6 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ 𝑧𝑆𝑤))
35	5, 34	syl5bb 282	. . . . 5 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
36	35	adantl 481	. . . 4 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
37	36	eqbrrdv2 36804	. . 3 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
38	4, 37	mpanl1 696	. 2 ⊢ ((Rel 𝑆 ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
39	2, 38	mpancom 684	1 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880  ∅c0 4253  {csn 4558  ∪ cuni 4836   class class class wbr 5070  {copab 5132  Rel wrel 5585   Er wer 8453  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-er 8456  df-ec 8458  df-qs 8462

This theorem is referenced by: (None)