Theorem prter2 38480

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 3275	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
2		r19.41v 3178	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
3	2	exbii 1842	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
4	1, 3	bitri 274	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
5		df-rex 3060	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
6	5	rexbii 3083	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
7		vex 3465	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
8	7	elqs 8788	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ )
9		df-rex 3060	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ))
10		eluni2 4913	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
11	10	anbi1i 622	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
12	11	exbii 1842	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
13	9, 12	bitri 274	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
14	8, 13	bitri 274	. . . . . . . . . 10 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
15	4, 6, 14	3bitr4ri 303	. . . . . . . . 9 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ )
16		prtlem18.1	. . . . . . . . . . . 12 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
17	16	prtlem19 38477	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
18	17	ralrimivv 3188	. . . . . . . . . 10 ⊢ (Prt 𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ )
19		2r19.29 3128	. . . . . . . . . . 11 ⊢ ((∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ))
20	19	ex 411	. . . . . . . . . 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	biimtrid 241	. . . . . . . 8 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )))
23		eqtr3 2751	. . . . . . . . . 10 ⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝)
24	23	reximi 3073	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)
25	24	reximi 3073	. . . . . . . 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 3060	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
28		19.41v 1945	. . . . . . . . . 10 ⊢ (∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝) ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
29	27, 28	bitri 274	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
30	29	simprbi 495	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝)
31	30	reximi 3073	. . . . . . 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 3220	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝)
34	32, 33	imbitrrdi 251	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ 𝐴))
35	16	prtlem400 38469	. . . . . 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 511	. . . 4 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)))
39		eldifsn 4792	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))
40	38, 39	imbitrrdi 251	. . 3 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41		neldifsn 4797	. . . . . . 7 ⊢ ¬ ∅ ∈ (𝐴 ∖ {∅})
42		n0el 4361	. . . . . . 7 ⊢ (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝)
43	41, 42	mpbi 229	. . . . . 6 ⊢ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝
44	43	rspec 3237	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝)
45		eldifi 4123	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴)
46	44, 45	jca 510	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
47	16	prtlem19 38477	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ ))
48	47	ancomsd 464	. . . . . . . 8 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ ))
49		elunii 4914	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
50	48, 49	jca2r 38454	. . . . . . 7 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )))
51		prtlem11 38465	. . . . . . . . 9 ⊢ (𝑝 ∈ V → (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ ))))
52	51	elv 3467	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝐴 → (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴 / ∼ )))
53	52	imp 405	. . . . . . 7 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴 / ∼ ))
54	50, 53	syl6 35	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
55	54	eximdv 1912	. . . . 5 ⊢ (Prt 𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ )))
56		19.41v 1945	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
57		19.9v 1979	. . . . 5 ⊢ (∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (∪ 𝐴 / ∼ ))
58	55, 56, 57	3imtr3g 294	. . . 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 211	. 2 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ 𝑝 ∈ (𝐴 ∖ {∅})))
61	60	eqrdv 2723	1 ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ∖ cdif 3941  ∅c0 4322  {csn 4630  ∪ cuni 4909  {copab 5211  [cec 8723   / cqs 8724  Prt wprt 38470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731  df-prt 38471

This theorem is referenced by: (None)