Theorem prter2 34689

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 3377	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ ∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
2		r19.41v 3237	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
3	2	exbii 1924	. . . . . . . . . . 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 3067	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
6	5	rexbii 3189	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
7		vex 3354	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
8	7	elqs 7951	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (∪ 𝐴 / ∼ ) ↔ ∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ )
9		df-rex 3067	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ))
10		eluni2 4578	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣)
11	10	anbi1i 610	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) ↔ (∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ))
12	11	exbii 1924	. . . . . . . . . . . 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 34686	. . . . . . . . . . 11 ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
18	17	ralrimivv 3119	. . . . . . . . . 10 ⊢ (Prt 𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ )
19		2r19.29 3227	. . . . . . . . . . 11 ⊢ ((∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ))
20	19	ex 397	. . . . . . . . . 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 2792	. . . . . . . . . 10 ⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝)
24	23	reximi 3159	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)
25	24	reximi 3159	. . . . . . . 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 3067	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝))
28		19.41v 2029	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝)
31	30	reximi 3159	. . . . . . 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 3210	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝)
34	32, 33	syl6ibr 242	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ 𝐴))
35	16	prtlem400 34678	. . . . . 6 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )
36		nelelne 3041	. . . . . 6 ⊢ (¬ ∅ ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
37	35, 36	mp1i 13	. . . . 5 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ≠ ∅))
38	34, 37	jcad 502	. . . 4 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)))
39		eldifsn 4453	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))
40	38, 39	syl6ibr 242	. . 3 ⊢ (Prt 𝐴 → (𝑝 ∈ (∪ 𝐴 / ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅})))
41		neldifsn 4458	. . . . . . 7 ⊢ ¬ ∅ ∈ (𝐴 ∖ {∅})
42		n0el 4087	. . . . . . 7 ⊢ (¬ ∅ ∈ (𝐴 ∖ {∅}) ↔ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝)
43	41, 42	mpbi 220	. . . . . 6 ⊢ ∀𝑝 ∈ (𝐴 ∖ {∅})∃𝑧 𝑧 ∈ 𝑝
44	43	rspec 3080	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝)
45		eldifi 3883	. . . . 5 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴)
46	44, 45	jca 501	. . . 4 ⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
47	16	prtlem19 34686	. . . . . . . . 9 ⊢ (Prt 𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ ))
48	47	ancomsd 456	. . . . . . . 8 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ ))
49		elunii 4579	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
50	48, 49	jca2r 34662	. . . . . . 7 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )))
51		prtlem11 34674	. . . . . . . . 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 393	. . . . . . 7 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴 / ∼ ))
54	50, 53	syl6 35	. . . . . 6 ⊢ (Prt 𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼ )))
55	54	eximdv 1998	. . . . 5 ⊢ (Prt 𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼ )))
56		19.41v 2029	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴))
57		19.9v 2065	. . . . 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 2769	1 ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∖ cdif 3720  ∅c0 4063  {csn 4316  ∪ cuni 4574  {copab 4846  [cec 7894   / cqs 7895  Prt wprt 34679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898  df-qs 7902  df-prt 34680

This theorem is referenced by: (None)