Theorem prter2 36581

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 3162	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
2		r19.41v 3250	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
3	2	exbii 1855	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
4	1, 3	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
5		df-rex 3057	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
6	5	rexbii 3160	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
7		vex 3402	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
8	7	elqs 8429	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ )
9		df-rex 3057	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ))
10		eluni2 4809	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
11	10	anbi1i 627	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
12	11	exbii 1855	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
13	9, 12	bitri 278	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
14	8, 13	bitri 278	. . . . . . . . . 10 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
15	4, 6, 14	3bitr4ri 307	. . . . . . . . 9 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ )
16		prtlem18.1	. . . . . . . . . . . 12 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
17	16	prtlem19 36578	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
18	17	ralrimivv 3101	. . . . . . . . . 10 ⊢ (Prt 𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ )
19		2r19.29 3240	. . . . . . . . . . 11 ⊢ ((∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ))
20	19	ex 416	. . . . . . . . . 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 245	. . . . . . . 8 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )))
23		eqtr3 2758	. . . . . . . . . 10 ⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝)
24	23	reximi 3156	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)
25	24	reximi 3156	. . . . . . . 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 3057	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
28		19.41v 1958	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝) ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
29	27, 28	bitri 278	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
30	29	simprbi 500	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝)
31	30	reximi 3156	. . . . . . 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 3176	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝)
34	32, 33	syl6ibr 255	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ 𝐴))
35	16	prtlem400 36570	. . . . . 6 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )
36		nelelne 3030	. . . . . 6 ⊢ (¬ ∅ ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
37	35, 36	mp1i 13	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
38	34, 37	jcad 516	. . . 4 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)))
39		eldifsn 4686	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))
40	38, 39	syl6ibr 255	. . 3 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41		neldifsn 4691	. . . . . . 7 ⊢ ¬ ∅ ∈ (𝐴 ∖ {∅})
42		n0el 4262	. . . . . . 7 ⊢ (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝)
43	41, 42	mpbi 233	. . . . . 6 ⊢ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝
44	43	rspec 3119	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝)
45		eldifi 4027	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴)
46	44, 45	jca 515	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
47	16	prtlem19 36578	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ ))
48	47	ancomsd 469	. . . . . . . 8 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ ))
49		elunii 4810	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
50	48, 49	jca2r 36555	. . . . . . 7 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )))
51		prtlem11 36566	. . . . . . . . 9 ⊢ (𝑝 ∈ V → (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ ))))
52	51	elv 3404	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ )))
53	52	imp 410	. . . . . . 7 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴 / ∼ ))
54	50, 53	syl6 35	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
55	54	eximdv 1925	. . . . 5 ⊢ (Prt 𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ )))
56		19.41v 1958	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
57		19.9v 1993	. . . . 5 ⊢ (∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (∪ 𝐴 / ∼ ))
58	55, 56, 57	3imtr3g 298	. . . 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 215	. 2 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
61	60	eqrdv 2734	1 ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051  ∃wrex 3052  Vcvv 3398   ∖ cdif 3850  ∅c0 4223  {csn 4527  ∪ cuni 4805  {copab 5101  [cec 8367   / cqs 8368  Prt wprt 36571

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-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ec 8371  df-qs 8375  df-prt 36572

This theorem is referenced by: (None)