Theorem 1stckgenlem 23576

Description: The one-point compactification of ℕ is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypotheses

Ref	Expression
1stckgen.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
1stckgen.2	⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
1stckgen.3	⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)

Assertion

Ref	Expression
1stckgenlem	⊢ (𝜑 → (𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp)

Proof of Theorem 1stckgenlem

Dummy variables 𝑗 𝑘 𝑛 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
2		ssun2 4188	. . . . . . . . 9 ⊢ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3		1stckgen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		1stckgen.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)
5		lmcl 23320	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝐴) → 𝐴 ∈ 𝑋)
6	3, 4, 5	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7		snssg 4787	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
9	2, 8	mpbiri 258	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
11	1, 10	sseldd 3995	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ ∪ 𝑢)
12		eluni2 4915	. . . . . 6 ⊢ (𝐴 ∈ ∪ 𝑢 ↔ ∃𝑤 ∈ 𝑢 𝐴 ∈ 𝑤)
13	11, 12	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑤 ∈ 𝑢 𝐴 ∈ 𝑤)
14		nnuz 12918	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
15		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐴 ∈ 𝑤)
16		1zzd 12645	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 1 ∈ ℤ)
17	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐹(⇝𝑡‘𝐽)𝐴)
18		simplrl 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ∈ 𝒫 𝐽)
19	18	elpwid 4613	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ⊆ 𝐽)
20		simprl 771	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝑢)
21	19, 20	sseldd 3995	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝐽)
22	14, 15, 16, 17, 21	lmcvg 23285	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
23		imassrn 6090	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24		ssun1 4187	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
25	23, 24	sstri 4004	. . . . . . . . . . . 12 ⊢ (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26		id 22	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
27	25, 26	sstrid 4006	. . . . . . . . . . 11 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢)
28		1stckgen.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
29	28	frnd 6744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
30	23, 29	sstrid 4006	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31		resttopon 23184	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
32	3, 30, 31	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
33		topontop 22934	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Top)
34	32, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Top)
35		fzfid 14010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑗) ∈ Fin)
36	28	ffund 6740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐹)
37		fz1ssnn 13591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑗) ⊆ ℕ
38	28	fdmd 6746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = ℕ)
39	37, 38	sseqtrrid 4048	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40		fores 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
41	36, 39, 40	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
42		fofi 9348	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
43	35, 41, 42	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
44		pwfi 9354	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
45	43, 44	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46		restsspw 17477	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ↾t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47		ssfi 9211	. . . . . . . . . . . . . . . 16 ⊢ ((𝒫 (𝐹 “ (1...𝑗)) ∈ Fin ∧ (𝐽 ↾t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Fin)
48	45, 46, 47	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Fin)
49	34, 48	elind 4209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50		fincmp 23416	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp)
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp)
52		topontop 22934	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
53	3, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ Top)
54		toponuni 22935	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
55	3, 54	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
56	30, 55	sseqtrd 4035	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝐽)
57		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
58	57	cmpsub 23423	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝐽) → ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)))
59	53, 56, 58	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)))
60	51, 59	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
61	60	r19.21bi 3248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
62	27, 61	syl5 34	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
63	62	impr 454	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)
64	63	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)
65		simprl 771	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑠 ∈ (𝒫 𝑢 ∩ Fin))
66	65	elin1d 4213	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑠 ∈ 𝒫 𝑢)
67	66	elpwid 4613	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑠 ⊆ 𝑢)
68		simprll 779	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) → 𝑤 ∈ 𝑢)
69	68	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑤 ∈ 𝑢)
70	69	snssd 4813	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → {𝑤} ⊆ 𝑢)
71	67, 70	unssd 4201	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (𝑠 ∪ {𝑤}) ⊆ 𝑢)
72		vex 3481	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
73	72	elpw2 5339	. . . . . . . . . . 11 ⊢ ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑢)
74	71, 73	sylibr 234	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢)
75	65	elin2d 4214	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑠 ∈ Fin)
76		snfi 9081	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
77		unfi 9209	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
78	75, 76, 77	sylancl 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ Fin)
79	74, 78	elind 4209	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin))
80	28	ffnd 6737	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℕ)
81	80	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝐹 Fn ℕ)
82		simprrr 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
83	82	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
84		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑤 ↔ (𝐹‘𝑛) ∈ 𝑤))
86	85	rspccva 3620	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑛) ∈ 𝑤)
87	83, 86	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑛) ∈ 𝑤)
88		elun2 4192	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ 𝑤 → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
89	87, 88	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
90	89	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
91		elnnuz 12919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
92	91	anbi1i 624	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
93		elfzuzb 13554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
94	92, 93	bitr4i 278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ 𝑛 ∈ (1...𝑗))
95		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)
96		funimass4 6972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠))
97	36, 39, 96	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠))
98	97	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠))
99	95, 98	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠)
100	99	r19.21bi 3248	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ (1...𝑗)) → (𝐹‘𝑛) ∈ ∪ 𝑠)
101		elun1 4191	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ ∪ 𝑠 → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
102	100, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ (1...𝑗)) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
103	94, 102	sylan2b 594	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ (𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛))) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
104	103	anassrs 467	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
105		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
106	105	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝑗 ∈ ℕ)
107		nnz 12631	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108		nnz 12631	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109		uztric 12899	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ≥‘𝑗) ∨ 𝑗 ∈ (ℤ≥‘𝑛)))
110	107, 108, 109	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑗) ∨ 𝑗 ∈ (ℤ≥‘𝑛)))
111	106, 110	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑗) ∨ 𝑗 ∈ (ℤ≥‘𝑛)))
112	90, 104, 111	mpjaodan 960	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
113	112	ralrimiva 3143	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
114		fnfvrnss 7140	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤)) → ran 𝐹 ⊆ (∪ 𝑠 ∪ 𝑤))
115	81, 113, 114	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ran 𝐹 ⊆ (∪ 𝑠 ∪ 𝑤))
116		elun2 4192	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑤 → 𝐴 ∈ (∪ 𝑠 ∪ 𝑤))
117	116	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)) → 𝐴 ∈ (∪ 𝑠 ∪ 𝑤))
118	117	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → 𝐴 ∈ (∪ 𝑠 ∪ 𝑤))
119	118	snssd 4813	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → {𝐴} ⊆ (∪ 𝑠 ∪ 𝑤))
120	115, 119	unssd 4201	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (ran 𝐹 ∪ {𝐴}) ⊆ (∪ 𝑠 ∪ 𝑤))
121		uniun 4934	. . . . . . . . . . 11 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ ∪ {𝑤})
122		unisnv 4931	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
123	122	uneq2i 4174	. . . . . . . . . . 11 ⊢ (∪ 𝑠 ∪ ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
124	121, 123	eqtri 2762	. . . . . . . . . 10 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
125	120, 124	sseqtrrdi 4046	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤}))
126		unieq 4922	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ∪ 𝑣 = ∪ (𝑠 ∪ {𝑤}))
127	126	sseq2d 4027	. . . . . . . . . 10 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})))
128	127	rspcev 3621	. . . . . . . . 9 ⊢ (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
129	79, 125, 128	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
130	64, 129	rexlimddv 3158	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ ((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
131	130	anassrs 467	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
132	22, 131	rexlimddv 3158	. . . . 5 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
133	13, 132	rexlimddv 3158	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
134	133	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
135	134	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
136	6	snssd 4813	. . . . 5 ⊢ (𝜑 → {𝐴} ⊆ 𝑋)
137	29, 136	unssd 4201	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
138	137, 55	sseqtrd 4035	. . 3 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝐽)
139	57	cmpsub 23423	. . 3 ⊢ ((𝐽 ∈ Top ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝐽) → ((𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)))
140	53, 138, 139	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)))
141	135, 140	mpbird 257	1 ⊢ (𝜑 → (𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ∪ cun 3960   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  {csn 4630  ∪ cuni 4911   class class class wbr 5147  dom cdm 5688  ran crn 5689   ↾ cres 5690   “ cima 5691  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  –onto→wfo 6560  ‘cfv 6562  (class class class)co 7430  Fincfn 8983  1c1 11153  ℕcn 12263  ℤcz 12610  ℤ_≥cuz 12875  ...cfz 13543   ↾_t crest 17466  Topctop 22914  TopOnctopon 22931  ⇝_𝑡clm 23249  Compccmp 23409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-rest 17468  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-lm 23252  df-cmp 23410

This theorem is referenced by:  1stckgen  23577