sigapildsys - Mathbox for Thierry Arnoux

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Theorem sigapildsys 33160

Description: Sigma-algebra are exactly classes which are both lambda and pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)

**Hypotheses**
Ref	Expression
dynkin.p	⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
dynkin.l	⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}

**Assertion**
Ref	Expression
sigapildsys	⊢ (sigAlgebra‘𝑂) = (𝑃 ∩ 𝐿)

Distinct variable groups: 𝑥,𝑠,𝑦 𝑥,𝐿,𝑦 𝑂,𝑠,𝑥 𝑥,𝑃,𝑦 Allowed substitution hints: 𝑃(𝑠) 𝐿(𝑠) 𝑂(𝑦)

Proof of Theorem sigapildsys Dummy variables 𝑓 𝑖 𝑛 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dynkin.p	. . . 4 ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
2	1	sigapisys 33153	. . 3 ⊢ (sigAlgebra‘𝑂) ⊆ 𝑃
3		dynkin.l	. . . 4 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
4	3	sigaldsys 33157	. . 3 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿
5	2, 4	ssini 4232	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ (𝑃 ∩ 𝐿)
6		id 22	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ (𝑃 ∩ 𝐿))
7	6	elin1d 4199	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝑃)
8	1	ispisys 33150	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑃 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑡) ⊆ 𝑡))
9	7, 8	sylib 217	. . . . . . 7 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑡) ⊆ 𝑡))
10	9	simpld 496	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝒫 𝒫 𝑂)
11	10	elpwid 4612	. . . . 5 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ⊆ 𝒫 𝑂)
12		dif0 4373	. . . . . . 7 ⊢ (𝑂 ∖ ∅) = 𝑂
13	6	elin2d 4200	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝐿)
14	3	isldsys 33154	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))))
15	13, 14	sylib 217	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))))
16	15	simprd 497	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)))
17	16	simp2d 1144	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
18	16	simp1d 1143	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∅ ∈ 𝑡)
19		difeq2 4117	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑂 ∖ 𝑥) = (𝑂 ∖ ∅))
20		eqidd 2734	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → 𝑡 = 𝑡)
21	19, 20	eleq12d 2828	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑂 ∖ 𝑥) ∈ 𝑡 ↔ (𝑂 ∖ ∅) ∈ 𝑡))
22	21	rspcv 3609	. . . . . . . . 9 ⊢ (∅ ∈ 𝑡 → (∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 → (𝑂 ∖ ∅) ∈ 𝑡))
23	18, 22	syl 17	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 → (𝑂 ∖ ∅) ∈ 𝑡))
24	17, 23	mpd 15	. . . . . . 7 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑂 ∖ ∅) ∈ 𝑡)
25	12, 24	eqeltrrid 2839	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑂 ∈ 𝑡)
26		unieq 4920	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
27		uni0 4940	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
28	26, 27	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
29	28	adantl 483	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∪ 𝑥 = ∅)
30	18	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∅ ∈ 𝑡)
31	29, 30	eqeltrd 2834	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∪ 𝑥 ∈ 𝑡)
32		vex 3479	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
33	32	0sdom 9107	. . . . . . . . . . . . 13 ⊢ (∅ ≺ 𝑥 ↔ 𝑥 ≠ ∅)
34	33	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ → ∅ ≺ 𝑥)
35	34	adantl 483	. . . . . . . . . . 11 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∅ ≺ 𝑥)
36		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → 𝑥 ≼ ω)
37		nnenom 13945	. . . . . . . . . . . . 13 ⊢ ℕ ≈ ω
38	37	ensymi 9000	. . . . . . . . . . . 12 ⊢ ω ≈ ℕ
39		domentr 9009	. . . . . . . . . . . 12 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ)
40	36, 38, 39	sylancl 587	. . . . . . . . . . 11 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → 𝑥 ≼ ℕ)
41		fodomr 9128	. . . . . . . . . . 11 ⊢ ((∅ ≺ 𝑥 ∧ 𝑥 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑥)
42	35, 40, 41	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑥)
43		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
44	43	iundisj 25065	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
45		fofn 6808	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝑥 → 𝑓 Fn ℕ)
46		fniunfv 7246	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ ran 𝑓)
47	45, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ ran 𝑓)
48		forn 6809	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝑥 → ran 𝑓 = 𝑥)
49	48	unieqd 4923	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ ran 𝑓 = ∪ 𝑥)
50	47, 49	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ 𝑥)
51	44, 50	eqtr3id 2787	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) = ∪ 𝑥)
52	51	adantl 483	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) = ∪ 𝑥)
53		fvex 6905	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑛) ∈ V
54		difexg 5328	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑛) ∈ V → ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) ∈ V)
55	53, 54	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) ∈ V
56	55	dfiun3 5966	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) = ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
57		nfv 1918	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥)
58		nfcv 2904	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛𝑦
59		nfmpt1 5257	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
60	59	nfrn 5952	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
61	58, 60	nfel 2918	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
62	57, 61	nfan 1903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))))
63		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
64		nfv 1918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥)
65		nfcv 2904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖𝑦
66		nfcv 2904	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑖ℕ
67		nfcv 2904	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑖(𝑓‘𝑛)
68		nfiu1 5032	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑖∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)
69	67, 68	nfdif 4126	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑖((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
70	66, 69	nfmpt 5256	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑖(𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
71	70	nfrn 5952	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
72	65, 71	nfel 2918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
73	64, 72	nfan 1903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖(((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))))
74		nfv 1918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖 𝑛 ∈ ℕ
75	73, 74	nfan 1903	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ)
76	65, 69	nfeq 2917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
77	75, 76	nfan 1903	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖(((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
78	6	ad7antr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑡 ∈ (𝑃 ∩ 𝐿))
79		simp-4r 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → 𝑥 ∈ 𝒫 𝑡)
80	79	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑥 ∈ 𝒫 𝑡)
81	80	elpwid 4612	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑥 ⊆ 𝑡)
82		fof 6806	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:ℕ–onto→𝑥 → 𝑓:ℕ⟶𝑥)
83	82	ad4antlr 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑓:ℕ⟶𝑥)
84		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑛 ∈ ℕ)
85	83, 84	ffvelcdmd 7088	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (𝑓‘𝑛) ∈ 𝑥)
86	81, 85	sseldd 3984	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (𝑓‘𝑛) ∈ 𝑡)
87		fzofi 13939	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1..^𝑛) ∈ Fin
88	87	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (1..^𝑛) ∈ Fin)
89	81	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∧ 𝑖 ∈ (1..^𝑛)) → 𝑥 ⊆ 𝑡)
90	83	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∧ 𝑖 ∈ (1..^𝑛)) → 𝑓:ℕ⟶𝑥)
91		fzossnn 13681	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1..^𝑛) ⊆ ℕ
92	91	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (1..^𝑛) ⊆ ℕ)
93	92	sselda 3983	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∧ 𝑖 ∈ (1..^𝑛)) → 𝑖 ∈ ℕ)
94	90, 93	ffvelcdmd 7088	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∧ 𝑖 ∈ (1..^𝑛)) → (𝑓‘𝑖) ∈ 𝑥)
95	89, 94	sseldd 3984	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∧ 𝑖 ∈ (1..^𝑛)) → (𝑓‘𝑖) ∈ 𝑡)
96	1, 3, 77, 78, 86, 88, 95	sigapildsyslem 33159	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) ∈ 𝑡)
97	63, 96	eqeltrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑦 ∈ 𝑡)
98		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) → 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))))
99		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) = (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
100	99, 55	elrnmpti 5960	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ↔ ∃𝑛 ∈ ℕ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
101	98, 100	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) → ∃𝑛 ∈ ℕ 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
102	62, 97, 101	r19.29af 3266	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))) → 𝑦 ∈ 𝑡)
103	102	ex 414	. . . . . . . . . . . . . . 15 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → 𝑦 ∈ 𝑡))
104	103	ssrdv 3989	. . . . . . . . . . . . . 14 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ⊆ 𝑡)
105		nnex 12218	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
106	105	mptex 7225	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ V
107	106	rnex 7903	. . . . . . . . . . . . . . 15 ⊢ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ V
108		elpwg 4606	. . . . . . . . . . . . . . 15 ⊢ (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ V → (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡 ↔ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ⊆ 𝑡))
109	107, 108	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡 ↔ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ⊆ 𝑡)
110	104, 109	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡)
111	16	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
112	111	ad4antr 731	. . . . . . . . . . . . 13 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
113		nnct 13946	. . . . . . . . . . . . . . 15 ⊢ ℕ ≼ ω
114		mptct 10533	. . . . . . . . . . . . . . 15 ⊢ (ℕ ≼ ω → (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω)
115	113, 114	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω
116		rnct 10520	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω → ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω)
117	115, 116	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω)
118	43	iundisj2 25066	. . . . . . . . . . . . . 14 ⊢ Disj 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
119		disjrnmpt 31816	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) → Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦)
120	118, 119	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦)
121		breq1 5152	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (𝑥 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω))
122		disjeq1 5121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦))
123	121, 122	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) ↔ (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω ∧ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦)))
124		unieq 4920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → ∪ 𝑥 = ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))))
125	124	eleq1d 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (∪ 𝑥 ∈ 𝑡 ↔ ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡))
126	123, 125	imbi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡) ↔ ((ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω ∧ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦) → ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡)))
127	126	rspcv 3609	. . . . . . . . . . . . . . 15 ⊢ (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡 → (∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡) → ((ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω ∧ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦) → ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡)))
128	127	imp 408	. . . . . . . . . . . . . 14 ⊢ ((ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)) → ((ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω ∧ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦) → ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡))
129	128	imp 408	. . . . . . . . . . . . 13 ⊢ (((ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝒫 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)) ∧ (ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ≼ ω ∧ Disj 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))𝑦)) → ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡)
130	110, 112, 117, 120, 129	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ 𝑡)
131	56, 130	eqeltrid 2838	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) ∈ 𝑡)
132	52, 131	eqeltrrd 2835	. . . . . . . . . 10 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∪ 𝑥 ∈ 𝑡)
133	42, 132	exlimddv 1939	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∪ 𝑥 ∈ 𝑡)
134	31, 133	pm2.61dane 3030	. . . . . . . 8 ⊢ (((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
135	134	ex 414	. . . . . . 7 ⊢ ((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))
136	135	ralrimiva 3147	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))
137	25, 17, 136	3jca 1129	. . . . 5 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡)))
138	11, 137	jca 513	. . . 4 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑡 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))))
139		vex 3479	. . . . 5 ⊢ 𝑡 ∈ V
140		issiga 33110	. . . . 5 ⊢ (𝑡 ∈ V → (𝑡 ∈ (sigAlgebra‘𝑂) ↔ (𝑡 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡)))))
141	139, 140	ax-mp 5	. . . 4 ⊢ (𝑡 ∈ (sigAlgebra‘𝑂) ↔ (𝑡 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))))
142	138, 141	sylibr 233	. . 3 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ (sigAlgebra‘𝑂))
143	142	ssriv 3987	. 2 ⊢ (𝑃 ∩ 𝐿) ⊆ (sigAlgebra‘𝑂)
144	5, 143	eqssi 3999	1 ⊢ (sigAlgebra‘𝑂) = (𝑃 ∩ 𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 Disj wdisj 5114 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 Fn wfn 6539 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 ωcom 7855 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 ficfi 9405 1c1 11111 ℕcn 12212 ..^cfzo 13627 sigAlgebracsiga 33106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-siga 33107

This theorem is referenced by: dynkin 33165

W3C validator