Theorem 1stckgenlem 23468

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 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
2		ssun2 4126	. . . . . . . . 9 ⊢ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3		1stckgen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		1stckgen.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)
5		lmcl 23212	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝐴) → 𝐴 ∈ 𝑋)
6	3, 4, 5	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7		snssg 4733	. . . . . . . . . 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 3930	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ ∪ 𝑢)
12		eluni2 4860	. . . . . 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 12775	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
15		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐴 ∈ 𝑤)
16		1zzd 12503	. . . . . . 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 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ∈ 𝒫 𝐽)
19	18	elpwid 4556	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑢 ⊆ 𝐽)
20		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝑢)
21	19, 20	sseldd 3930	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝐽)
22	14, 15, 16, 17, 21	lmcvg 23177	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
23		imassrn 6019	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24		ssun1 4125	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
25	23, 24	sstri 3939	. . . . . . . . . . . 12 ⊢ (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26		id 22	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
27	25, 26	sstrid 3941	. . . . . . . . . . 11 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢)
28		1stckgen.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
29	28	frnd 6659	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
30	23, 29	sstrid 3941	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31		resttopon 23076	. . . . . . . . . . . . . . . . 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 22828	. . . . . . . . . . . . . . . 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 13880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑗) ∈ Fin)
36	28	ffund 6655	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐹)
37		fz1ssnn 13455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑗) ⊆ ℕ
38	28	fdmd 6661	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = ℕ)
39	37, 38	sseqtrrid 3973	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40		fores 6745	. . . . . . . . . . . . . . . . . . 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 9197	. . . . . . . . . . . . . . . . . 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 9203	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
45	43, 44	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46		restsspw 17335	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ↾t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47		ssfi 9082	. . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50		fincmp 23308	. . . . . . . . . . . . . 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 22828	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
53	3, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ Top)
54		toponuni 22829	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
55	3, 54	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
56	30, 55	sseqtrd 3966	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝐽)
57		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
58	57	cmpsub 23315	. . . . . . . . . . . . . 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 3224	. . . . . . . . . . 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 770	. . . . . . . . . . . . . 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 4151	. . . . . . . . . . . . 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 4556	. . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 4758	. . . . . . . . . . . 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 4139	. . . . . . . . . . 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 3440	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
73	72	elpw2 5270	. . . . . . . . . . 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 4152	. . . . . . . . . . 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 8965	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
77		unfi 9080	. . . . . . . . . . 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 4147	. . . . . . . . 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 6652	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑤 ↔ (𝐹‘𝑛) ∈ 𝑤))
86	85	rspccva 3571	. . . . . . . . . . . . . . . . 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 4130	. . . . . . . . . . . . . . . 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 12776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
92	91	anbi1i 624	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
93		elfzuzb 13418	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . . . . . 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 3224	. . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 12489	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108		nnz 12489	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109		uztric 12756	. . . . . . . . . . . . . . . 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 3124	. . . . . . . . . . . 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 7054	. . . . . . . . . . . 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 4130	. . . . . . . . . . . . . 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 4758	. . . . . . . . . . 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 4139	. . . . . . . . . 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 4879	. . . . . . . . . . 11 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ ∪ {𝑤})
122		unisnv 4876	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
123	122	uneq2i 4112	. . . . . . . . . . 11 ⊢ (∪ 𝑠 ∪ ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
124	121, 123	eqtri 2754	. . . . . . . . . 10 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
125	120, 124	sseqtrrdi 3971	. . . . . . . . 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 4867	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ∪ 𝑣 = ∪ (𝑠 ∪ {𝑤}))
127	126	sseq2d 3962	. . . . . . . . . 10 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})))
128	127	rspcev 3572	. . . . . . . . 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 3139	. . . . . . 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 3139	. . . . 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 3139	. . . 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 3124	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
136	6	snssd 4758	. . . . 5 ⊢ (𝜑 → {𝐴} ⊆ 𝑋)
137	29, 136	unssd 4139	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
138	137, 55	sseqtrd 3966	. . 3 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝐽)
139	57	cmpsub 23315	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856   class class class wbr 5089  dom cdm 5614  ran crn 5615   ↾ cres 5616   “ cima 5617  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  1c1 11007  ℕcn 12125  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407   ↾_t crest 17324  Topctop 22808  TopOnctopon 22825  ⇝_𝑡clm 23141  Compccmp 23301

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fi 9295  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-rest 17326  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-lm 23144  df-cmp 23302

This theorem is referenced by:  1stckgen  23469