Theorem prter2 34166

Description: The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)

Hypothesis

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

Assertion

Ref	Expression
prter2	⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

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

Proof of Theorem prter2

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

Step	Hyp	Ref	Expression
1		rexcom4 3225	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
2		r19.41v 3089	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
3	2	exbii 1774	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
4	1, 3	bitri 264	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
5		df-rex 2918	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
6	5	rexbii 3041	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
7		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
8	7	elqs 7799	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ )
9		df-rex 2918	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ))
10		eluni2 4440	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
11	10	anbi1i 731	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
12	11	exbii 1774	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
13	9, 12	bitri 264	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
14	8, 13	bitri 264	. . . . . . . . . 10 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
15	4, 6, 14	3bitr4ri 293	. . . . . . . . 9 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ )
16		prtlem18.1	. . . . . . . . . . . 12 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
17	16	prtlem19 34163	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
18	17	ralrimivv 2970	. . . . . . . . . 10 ⊢ (Prt 𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ )
19		2r19.29 3079	. . . . . . . . . . 11 ⊢ ((∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ))
20	19	ex 450	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ → (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )))
21	18, 20	syl 17	. . . . . . . . 9 ⊢ (Prt 𝐴 → (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )))
22	15, 21	syl5bi 232	. . . . . . . 8 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )))
23		eqtr3 2643	. . . . . . . . . 10 ⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝)
24	23	reximi 3011	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)
25	24	reximi 3011	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)
26	22, 25	syl6 35	. . . . . . 7 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝))
27		df-rex 2918	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
28		19.41v 1914	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝) ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
29	27, 28	bitri 264	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
30	29	simprbi 480	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝)
31	30	reximi 3011	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → ∃𝑣 ∈ 𝐴 𝑣 = 𝑝)
32	26, 31	syl6 35	. . . . . 6 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → ∃𝑣 ∈ 𝐴 𝑣 = 𝑝))
33		risset 3062	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝)
34	32, 33	syl6ibr 242	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ 𝐴))
35	16	prtlem400 34155	. . . . . 6 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )
36		nelelne 2892	. . . . . 6 ⊢ (¬ ∅ ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
37	35, 36	mp1i 13	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
38	34, 37	jcad 555	. . . 4 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)))
39		eldifsn 4317	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))
40	38, 39	syl6ibr 242	. . 3 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41		neldifsn 4321	. . . . . . 7 ⊢ ¬ ∅ ∈ (𝐴 ∖ {∅})
42		n0el 3940	. . . . . . 7 ⊢ (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝)
43	41, 42	mpbi 220	. . . . . 6 ⊢ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝
44	43	rspec 2931	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝)
45		eldifi 3732	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴)
46	44, 45	jca 554	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
47	16	prtlem19 34163	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ ))
48	47	ancomsd 470	. . . . . . . 8 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ ))
49		elunii 4441	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
50	48, 49	jca2r 34139	. . . . . . 7 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )))
51		prtlem11 34151	. . . . . . . . 9 ⊢ (𝑝 ∈ V → (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ ))))
52	7, 51	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ )))
53	52	imp 445	. . . . . . 7 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴 / ∼ ))
54	50, 53	syl6 35	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
55	54	eximdv 1846	. . . . 5 ⊢ (Prt 𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ )))
56		19.41v 1914	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
57		19.9v 1896	. . . . 5 ⊢ (∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (∪ 𝐴 / ∼ ))
58	55, 56, 57	3imtr3g 284	. . . 4 ⊢ (Prt 𝐴 → ((∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
59	46, 58	syl5 34	. . 3 ⊢ (Prt 𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
60	40, 59	impbid 202	. 2 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
61	60	eqrdv 2620	1 ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  {csn 4177  ∪ cuni 4436  {copab 4712  [cec 7740   / cqs 7741  Prt wprt 34156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748  df-prt 34157

This theorem is referenced by: (None)