Theorem 1stckgenlem 23780

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 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
2		ssun2 4125	. . . . . . . . 9 ⊢ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3		1stckgen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		1stckgen.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)
5		lmcl 23523	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝐴) → 𝐴 ∈ 𝑋)
6	3, 4, 5	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7		snssg 4744	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
8	6, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
9	2, 8	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
11	1, 10	sseldd 3932	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ ∪ 𝑢)
12		eluni2 4871	. . . . . 6 ⊢ (𝐴 ∈ ∪ 𝑢 ↔ ∃𝑤 ∈ 𝑢 𝐴 ∈ 𝑤)
13	11, 12	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑤 ∈ 𝑢 𝐴 ∈ 𝑤)
14		nnuz 12927	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
15		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐴 ∈ 𝑤)
16		1zzd 12650	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 1 ∈ ℤ)
17	4	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐹(⇝𝑡‘𝐽)𝐴)
18		simplrl 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ∈ 𝒫 𝐽)
19	18	elpwid 4566	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ⊆ 𝐽)
20		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝑢)
21	19, 20	sseldd 3932	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝐽)
22	14, 15, 16, 17, 21	lmcvg 23488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
23		imassrn 6067	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24		ssun1 4124	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
25	23, 24	sstri 3940	. . . . . . . . . . . 12 ⊢ (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26		id 23	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
27	25, 26	sstrid 3942	. . . . . . . . . . 11 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢)
28		1stckgen.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
29	28	frnd 6712	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
30	23, 29	sstrid 3942	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31		resttopon 23387	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
32	3, 30, 31	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
33		topontop 23139	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Top)
34	32, 33	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Top)
35		fzfid 14038	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑗) ∈ Fin)
36	28	ffund 6708	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐹)
37		fz1ssnn 13611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑗) ⊆ ℕ
38	28	fdmd 6714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = ℕ)
39	37, 38	sseqtrrid 3974	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40		fores 6800	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
41	36, 39, 40	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
42		fofi 9284	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
43	35, 41, 42	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
44		pwfi 9289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
45	43, 44	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46		restsspw 17517	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ↾t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47		ssfi 9168	. . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Fin)
49	34, 48	elind 4146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50		fincmp 23619	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp)
51	49, 50	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ Comp)
52		topontop 23139	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
53	3, 52	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ Top)
54		toponuni 23140	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
55	3, 54	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
56	30, 55	sseqtrd 3967	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝐽)
57		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
58	57	cmpsub 23626	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
61	60	r19.21bi 3254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
62	27, 61	syl5 35	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠))
63	62	impr 460	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠)
64	63	adantr 486	. . . . . . . 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 783	. . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . 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 4566	. . . . . . . . . . . 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 791	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 4747	. . . . . . . . . . . 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 4138	. . . . . . . . . . 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 3454	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
73	72	elpw2 5299	. . . . . . . . . . 11 ⊢ ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑢)
74	71, 73	sylibr 237	. . . . . . . . . 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 4151	. . . . . . . . . . 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 9051	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
77		unfi 9166	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
78	75, 76, 77	sylancl 598	. . . . . . . . . 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 4146	. . . . . . . . 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 6704	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℕ)
81	80	ad3antrrr 743	. . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑤 ↔ (𝐹‘𝑛) ∈ 𝑤))
86	85	rspccva 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑛) ∈ 𝑤)
87	83, 86	sylan 592	. . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ 𝑤 → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
89	87, 88	syl 18	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 12928	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
92	91	anbi1i 636	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
93		elfzuzb 13573	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
94	92, 93	bitr4i 281	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ 𝑛 ∈ (1...𝑗))
95		simprr 785	. . . . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠))
97	36, 39, 96	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ ∪ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹‘𝑛) ∈ ∪ 𝑠))
98	97	ad3antrrr 743	. . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . 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 3254	. . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ ∪ 𝑠 → (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤))
102	100, 101	syl 18	. . . . . . . . . . . . . . . 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 606	. . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
106	105	ad2antlr 740	. . . . . . . . . . . . . . 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 12637	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108		nnz 12637	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109		uztric 12912	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ≥‘𝑗) ∨ 𝑗 ∈ (ℤ≥‘𝑛)))
110	107, 108, 109	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑗) ∨ 𝑗 ∈ (ℤ≥‘𝑛)))
111	106, 110	sylan 592	. . . . . . . . . . . . . 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 973	. . . . . . . . . . . . 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 3154	. . . . . . . . . . . 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 7115	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (∪ 𝑠 ∪ 𝑤)) → ran 𝐹 ⊆ (∪ 𝑠 ∪ 𝑤))
115	81, 113, 114	syl2anc 596	. . . . . . . . . . 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 4129	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑤 → 𝐴 ∈ (∪ 𝑠 ∪ 𝑤))
117	116	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)) → 𝐴 ∈ (∪ 𝑠 ∪ 𝑤))
118	117	ad2antlr 740	. . . . . . . . . . . 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 4747	. . . . . . . . . . 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 4138	. . . . . . . . . 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 4890	. . . . . . . . . . 11 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ ∪ {𝑤})
122		unisnv 4887	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
123	122	uneq2i 4112	. . . . . . . . . . 11 ⊢ (∪ 𝑠 ∪ ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
124	121, 123	eqtri 2783	. . . . . . . . . 10 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
125	120, 124	sseqtrrdi 3972	. . . . . . . . 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 4878	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ∪ 𝑣 = ∪ (𝑠 ∪ {𝑤}))
127	126	sseq2d 3963	. . . . . . . . . 10 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})))
128	127	rspcev 3576	. . . . . . . . 9 ⊢ (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
129	79, 125, 128	syl2anc 596	. . . . . . . 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 3169	. . . . . . 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 473	. . . . . 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 3169	. . . . 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 3169	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣)
134	133	expr 462	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
135	134	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
136	6	snssd 4747	. . . . 5 ⊢ (𝜑 → {𝐴} ⊆ 𝑋)
137	29, 136	unssd 4138	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
138	137, 55	sseqtrd 3967	. . 3 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝐽)
139	57	cmpsub 23626	. . 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 596	. 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 260	1 ⊢ (𝜑 → (𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   class class class wbr 5103  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7414  Fincfn 8953  1c1 11126  ℕcn 12258  ℤcz 12616  ℤ_≥cuz 12888  ...cfz 13562   ↾_t crest 17506  Topctop 23119  TopOnctopon 23136  ⇝_𝑡clm 23452  Compccmp 23612

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-pm 8830  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fi 9382  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-n0 12530  df-z 12617  df-uz 12889  df-fz 13563  df-rest 17508  df-topgen 17529  df-top 23120  df-topon 23137  df-bases 23172  df-lm 23455  df-cmp 23613

This theorem is used by:  1stckgen  23781