Theorem 1stckgenlem 23447

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 4145	. . . . . . . . 9 ⊢ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3		1stckgen.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		1stckgen.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)
5		lmcl 23191	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝐴) → 𝐴 ∈ 𝑋)
6	3, 4, 5	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7		snssg 4750	. . . . . . . . . 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 3950	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) → 𝐴 ∈ ∪ 𝑢)
12		eluni2 4878	. . . . . 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 12843	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
15		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝐴 ∈ 𝑤)
16		1zzd 12571	. . . . . . 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 4575	. . . . . . . 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 3950	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → 𝑤 ∈ 𝐽)
22	14, 15, 16, 17, 21	lmcvg 23156	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)) ∧ (𝑤 ∈ 𝑢 ∧ 𝐴 ∈ 𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑤)
23		imassrn 6045	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24		ssun1 4144	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
25	23, 24	sstri 3959	. . . . . . . . . . . 12 ⊢ (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26		id 22	. . . . . . . . . . . 12 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢)
27	25, 26	sstrid 3961	. . . . . . . . . . 11 ⊢ ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝑢)
28		1stckgen.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
29	28	frnd 6699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
30	23, 29	sstrid 3961	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31		resttopon 23055	. . . . . . . . . . . . . . . . 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 22807	. . . . . . . . . . . . . . . 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 13945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑗) ∈ Fin)
36	28	ffund 6695	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝐹)
37		fz1ssnn 13523	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑗) ⊆ ℕ
38	28	fdmd 6701	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = ℕ)
39	37, 38	sseqtrrid 3993	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40		fores 6785	. . . . . . . . . . . . . . . . . . 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 9269	. . . . . . . . . . . . . . . . . 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 9275	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
45	43, 44	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46		restsspw 17401	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ↾t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47		ssfi 9143	. . . . . . . . . . . . . . . 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 4166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 ↾t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50		fincmp 23287	. . . . . . . . . . . . . 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 22807	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
53	3, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ Top)
54		toponuni 22808	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
55	3, 54	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
56	30, 55	sseqtrd 3986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 “ (1...𝑗)) ⊆ ∪ 𝐽)
57		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
58	57	cmpsub 23294	. . . . . . . . . . . . . 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 3230	. . . . . . . . . . 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 4170	. . . . . . . . . . . . 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 4575	. . . . . . . . . . . 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 4776	. . . . . . . . . . . 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 4158	. . . . . . . . . . 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 5292	. . . . . . . . . . 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 4171	. . . . . . . . . . 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 9017	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
77		unfi 9141	. . . . . . . . . . 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 4166	. . . . . . . . 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 6692	. . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑤 ↔ (𝐹‘𝑛) ∈ 𝑤))
86	85	rspccva 3590	. . . . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . 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 12844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
92	91	anbi1i 624	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ↔ (𝑛 ∈ (ℤ≥‘1) ∧ 𝑗 ∈ (ℤ≥‘𝑛)))
93		elfzuzb 13486	. . . . . . . . . . . . . . . . 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 6928	. . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . . . 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 12557	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108		nnz 12557	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109		uztric 12824	. . . . . . . . . . . . . . . 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 3126	. . . . . . . . . . . 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 7096	. . . . . . . . . . . 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 4149	. . . . . . . . . . . . . 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 4776	. . . . . . . . . . 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 4158	. . . . . . . . . 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 4897	. . . . . . . . . . 11 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ ∪ {𝑤})
122		unisnv 4894	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
123	122	uneq2i 4131	. . . . . . . . . . 11 ⊢ (∪ 𝑠 ∪ ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
124	121, 123	eqtri 2753	. . . . . . . . . 10 ⊢ ∪ (𝑠 ∪ {𝑤}) = (∪ 𝑠 ∪ 𝑤)
125	120, 124	sseqtrrdi 3991	. . . . . . . . 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 4885	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ∪ 𝑣 = ∪ (𝑠 ∪ {𝑤}))
127	126	sseq2d 3982	. . . . . . . . . 10 ⊢ (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ ∪ (𝑠 ∪ {𝑤})))
128	127	rspcev 3591	. . . . . . . . 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 3141	. . . . . . 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 3141	. . . . 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 3141	. . . 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 3126	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝑣))
136	6	snssd 4776	. . . . 5 ⊢ (𝜑 → {𝐴} ⊆ 𝑋)
137	29, 136	unssd 4158	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
138	137, 55	sseqtrd 3986	. . 3 ⊢ (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ ∪ 𝐽)
139	57	cmpsub 23294	. . 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874   class class class wbr 5110  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  1c1 11076  ℕcn 12193  ℤcz 12536  ℤ_≥cuz 12800  ...cfz 13475   ↾_t crest 17390  Topctop 22787  TopOnctopon 22804  ⇝_𝑡clm 23120  Compccmp 23280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9369  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-rest 17392  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-lm 23123  df-cmp 23281

This theorem is referenced by:  1stckgen  23448