Theorem prter3 39939

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 8727	. . 3 ⊢ (𝑆 Er ∪ 𝐴 → Rel 𝑆)
2	1	adantr 486	. 2 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → Rel 𝑆)
3		prtlem18.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
4	3	relopabiv 5798	. . 3 ⊢ Rel ∼
5	3	prtlem13 39925	. . . . . 6 ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
6		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑆 Er ∪ 𝐴)
7		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ 𝐴)
8		ne0i 4287	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑣 → 𝑣 ≠ ∅)
9	8	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ≠ ∅)
10		eldifsn 4748	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴 ∖ {∅}) ↔ (𝑣 ∈ 𝐴 ∧ 𝑣 ≠ ∅))
11	7, 9, 10	sylanbrc 595	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (𝐴 ∖ {∅}))
12		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))
13	11, 12	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 ∈ (∪ 𝐴 / 𝑆))
14		simprr 785	. . . . . . . . . . . . 13 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑧 ∈ 𝑣)
15		qsel 8817	. . . . . . . . . . . . 13 ⊢ ((𝑆 Er ∪ 𝐴 ∧ 𝑣 ∈ (∪ 𝐴 / 𝑆) ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧]𝑆)
16	6, 13, 14, 15	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → 𝑣 = [𝑧]𝑆)
17	16	eleq2d 2847	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ (𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣)) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ [𝑧]𝑆))
18		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
19		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	18, 19	elec 8764	. . . . . . . . . . 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 473	. . . . . . . . 9 ⊢ ((((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧𝑆𝑤))
23	22	pm5.32da 590	. . . . . . . 8 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑣 ∈ 𝐴) → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
24	23	rexbidva 3185	. . . . . . 7 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
25		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑆 Er ∪ 𝐴)
26		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧𝑆𝑤)
27	25, 26	ercl 8729	. . . . . . . . . . 11 ⊢ (((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) ∧ 𝑧𝑆𝑤) → 𝑧 ∈ ∪ 𝐴)
28		eluni2 4871	. . . . . . . . . . 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 418	. . . . . . . . 9 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 → ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣))
31	30	pm4.71rd 572	. . . . . . . 8 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧𝑆𝑤 ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑧𝑆𝑤)))
32		r19.41v 3193	. . . . . . . 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	bitrid 286	. . . . 5 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
36	35	adantl 487	. . . 4 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → (𝑧 ∼ 𝑤 ↔ 𝑧𝑆𝑤))
37	36	eqbrrdv2 39920	. . 3 ⊢ (((Rel ∼ ∧ Rel 𝑆) ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
38	4, 37	mpanl1 713	. 2 ⊢ ((Rel 𝑆 ∧ (𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅}))) → ∼ = 𝑆)
39	2, 38	mpancom 701	1 ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896  ∅c0 4279  {csn 4584  ∪ cuni 4867   class class class wbr 5103  {copab 5167  Rel wrel 5656   Er wer 8714  [cec 8715   / cqs 8716

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-er 8717  df-ec 8719  df-qs 8723

This theorem is used by: (None)