Theorem sigapildsys 34196

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 34189	. . 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 34193	. . 3 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿
5	2, 4	ssini 4189	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ (𝑃 ∩ 𝐿)
6		id 22	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ (𝑃 ∩ 𝐿))
7	6	elin1d 4153	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝑃)
8	1	ispisys 34186	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑃 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑡) ⊆ 𝑡))
9	7, 8	sylib 218	. . . . . . 7 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑡) ⊆ 𝑡))
10	9	simpld 494	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝒫 𝒫 𝑂)
11	10	elpwid 4558	. . . . 5 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ⊆ 𝒫 𝑂)
12		dif0 4327	. . . . . . 7 ⊢ (𝑂 ∖ ∅) = 𝑂
13	6	elin2d 4154	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ 𝐿)
14	3	isldsys 34190	. . . . . . . . . . 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 218	. . . . . . . . . 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 495	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)))
17	16	simp2d 1143	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
18	16	simp1d 1142	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∅ ∈ 𝑡)
19		difeq2 4069	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑂 ∖ 𝑥) = (𝑂 ∖ ∅))
20		eqidd 2734	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → 𝑡 = 𝑡)
21	19, 20	eleq12d 2827	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑂 ∖ 𝑥) ∈ 𝑡 ↔ (𝑂 ∖ ∅) ∈ 𝑡))
22	21	rspcv 3569	. . . . . . . . 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 2838	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑂 ∈ 𝑡)
26		unieq 4869	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
27		uni0 4886	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
28	26, 27	eqtrdi 2784	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
29	28	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∪ 𝑥 = ∅)
30	18	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∅ ∈ 𝑡)
31	29, 30	eqeltrd 2833	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → ∪ 𝑥 ∈ 𝑡)
32		vex 3441	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
33	32	0sdom 9028	. . . . . . . . . . . . 13 ⊢ (∅ ≺ 𝑥 ↔ 𝑥 ≠ ∅)
34	33	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ ∅ → ∅ ≺ 𝑥)
35	34	adantl 481	. . . . . . . . . . 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 13889	. . . . . . . . . . . . 13 ⊢ ℕ ≈ ω
38	37	ensymi 8933	. . . . . . . . . . . 12 ⊢ ω ≈ ℕ
39		domentr 8942	. . . . . . . . . . . 12 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ)
40	36, 38, 39	sylancl 586	. . . . . . . . . . 11 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → 𝑥 ≼ ℕ)
41		fodomr 9048	. . . . . . . . . . 11 ⊢ ((∅ ≺ 𝑥 ∧ 𝑥 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑥)
42	35, 40, 41	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑥)
43		fveq2 6828	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
44	43	iundisj 25477	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
45		fofn 6742	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝑥 → 𝑓 Fn ℕ)
46		fniunfv 7187	. . . . . . . . . . . . . . 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 6743	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝑥 → ran 𝑓 = 𝑥)
49	48	unieqd 4871	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ ran 𝑓 = ∪ 𝑥)
50	47, 49	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) = ∪ 𝑥)
51	44, 50	eqtr3id 2782	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝑥 → ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) = ∪ 𝑥)
52	51	adantl 481	. . . . . . . . . . 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 6841	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑛) ∈ V
54		difexg 5269	. . . . . . . . . . . . . 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 5913	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)) = ∪ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
57		nfv 1915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥)
58		nfcv 2895	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛𝑦
59		nfmpt1 5192	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
60	59	nfrn 5896	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
61	58, 60	nfel 2910	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
62	57, 61	nfan 1900	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 1915	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥)
65		nfcv 2895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖𝑦
66		nfcv 2895	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑖ℕ
67		nfcv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑖(𝑓‘𝑛)
68		nfiu1 4977	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑖∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)
69	67, 68	nfdif 4078	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑖((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
70	66, 69	nfmpt 5191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑖(𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
71	70	nfrn 5896	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑖ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
72	65, 71	nfel 2910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑖 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖)))
73	64, 72	nfan 1900	. . . . . . . . . . . . . . . . . . . . 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 1915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑖 𝑛 ∈ ℕ
75	73, 74	nfan 1900	. . . . . . . . . . . . . . . . . . . 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 2909	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑖 𝑦 = ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
77	75, 76	nfan 1900	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 4558	. . . . . . . . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:ℕ–onto→𝑥 → 𝑓:ℕ⟶𝑥)
83	82	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . 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 13883	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 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 13613	. . . . . . . . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . 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 34195	. . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 5906	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . 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 3242	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 12138	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
106	105	mptex 7163	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ V
107	106	rnex 7846	. . . . . . . . . . . . . . 15 ⊢ ran (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))) ∈ V
108		elpwg 4552	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
112	111	ad4antr 732	. . . . . . . . . . . . 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 13890	. . . . . . . . . . . . . . 15 ⊢ ℕ ≼ ω
114		mptct 10436	. . . . . . . . . . . . . . 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 10423	. . . . . . . . . . . . . 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 25478	. . . . . . . . . . . . . 14 ⊢ Disj 𝑛 ∈ ℕ ((𝑓‘𝑛) ∖ ∪ 𝑖 ∈ (1..^𝑛)(𝑓‘𝑖))
119		disjrnmpt 32567	. . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . 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 5067	. . . . . . . . . . . . . . . . . 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 4869	. . . . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 2837	. . . . . . . . . . 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 2834	. . . . . . . . . 10 ⊢ (((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) ∧ 𝑓:ℕ–onto→𝑥) → ∪ 𝑥 ∈ 𝑡)
133	42, 132	exlimddv 1936	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∪ 𝑥 ∈ 𝑡)
134	31, 133	pm2.61dane 3016	. . . . . . . 8 ⊢ (((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑡)
135	134	ex 412	. . . . . . 7 ⊢ ((𝑡 ∈ (𝑃 ∩ 𝐿) ∧ 𝑥 ∈ 𝒫 𝑡) → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))
136	135	ralrimiva 3125	. . . . . 6 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))
137	25, 17, 136	3jca 1128	. . . . 5 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡)))
138	11, 137	jca 511	. . . 4 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → (𝑡 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑡))))
139		vex 3441	. . . . 5 ⊢ 𝑡 ∈ V
140		issiga 34146	. . . . 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 234	. . 3 ⊢ (𝑡 ∈ (𝑃 ∩ 𝐿) → 𝑡 ∈ (sigAlgebra‘𝑂))
143	142	ssriv 3934	. 2 ⊢ (𝑃 ∩ 𝐿) ⊆ (sigAlgebra‘𝑂)
144	5, 143	eqssi 3947	1 ⊢ (sigAlgebra‘𝑂) = (𝑃 ∩ 𝐿)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  {crab 3396  Vcvv 3437   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858  ∪ ciun 4941  Disj wdisj 5060   class class class wbr 5093   ↦ cmpt 5174  ran crn 5620   Fn wfn 6481  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7352  ωcom 7802   ≈ cen 8872   ≼ cdom 8873   ≺ csdm 8874  Fincfn 8875  ficfi 9301  1c1 11014  ℕcn 12132  ..^cfzo 13556  sigAlgebracsiga 34142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-ac2 10361  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-disj 5061  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fi 9302  df-sup 9333  df-inf 9334  df-oi 9403  df-dju 9801  df-card 9839  df-acn 9842  df-ac 10014  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-siga 34143

This theorem is referenced by:  dynkin  34201