Theorem 1stcfb 23818

Description: For any point 𝐴 in a first-countable topology, there is a function 𝑓:ℕ⟶𝐽 enumerating neighborhoods of 𝐴 which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypothesis

Ref	Expression
1stcclb.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
1stcfb	⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))

Distinct variable groups:   𝑓,𝑘,𝑦,𝐴   𝑓,𝐽,𝑘,𝑦   𝑘,𝑋,𝑦

Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem 1stcfb

Dummy variables 𝑎 𝑔 𝑛 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stcclb.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	1stcclb 23817	. 2 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
3		simplr 781	. . . . . . . . 9 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝐴 ∈ 𝑋)
4		eleq2 2853	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑋))
5		sseq2 3939	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ 𝑋))
6	5	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑋 → ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)))
7	6	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)))
8	4, 7	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → ((𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝐴 ∈ 𝑋 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋))))
9		simprrr 794	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
10		1stctop 23816	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
11	10	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝐽 ∈ Top)
12	1	topopn 23279	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑋 ∈ 𝐽)
14	8, 9, 13	rspcdva 3581	. . . . . . . . 9 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → (𝐴 ∈ 𝑋 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)))
15	3, 14	mpd 16	. . . . . . . 8 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋))
16		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋) → 𝐴 ∈ 𝑤)
17	16	reximi 3104	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋) → ∃𝑤 ∈ 𝑥 𝐴 ∈ 𝑤)
18	15, 17	syl 18	. . . . . . 7 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑤 ∈ 𝑥 𝐴 ∈ 𝑤)
19		eleq2w 2848	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑎))
20	19	cbvrexvw 3245	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝑥 𝐴 ∈ 𝑤 ↔ ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎)
21	18, 20	sylib 221	. . . . . 6 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎)
22		rabn0 4342	. . . . . 6 ⊢ ({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎)
23	21, 22	sylibr 237	. . . . 5 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅)
24		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
25	24	rabex 5307	. . . . . 6 ⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ∈ V
26	25	0sdom 9165	. . . . 5 ⊢ (∅ ≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ↔ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅)
27	23, 26	sylibr 237	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∅ ≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
28		ssrab2 4008	. . . . . 6 ⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝑥
29		ssdomg 9065	. . . . . 6 ⊢ (𝑥 ∈ V → ({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝑥 → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥))
30	24, 28, 29	mp2 9	. . . . 5 ⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥
31		simprrl 793	. . . . . 6 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑥 ≼ ω)
32		nnenom 14163	. . . . . . 7 ⊢ ℕ ≈ ω
33	32	ensymi 9069	. . . . . 6 ⊢ ω ≈ ℕ
34		domentr 9078	. . . . . 6 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ)
35	31, 33, 34	sylancl 598	. . . . 5 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑥 ≼ ℕ)
36		domtr 9072	. . . . 5 ⊢ (({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥 ∧ 𝑥 ≼ ℕ) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ)
37	30, 35, 36	sylancr 599	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ)
38		fodomr 9185	. . . 4 ⊢ ((∅ ≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ∧ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ) → ∃𝑔 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
39	27, 37, 38	syl2anc 596	. . 3 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑔 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
40	10	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Top)
41		imassrn 6216	. . . . . . . . . 10 ⊢ (𝑔 “ (1...𝑛)) ⊆ ran 𝑔
42		forn 6829	. . . . . . . . . . . . 13 ⊢ (𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
43	42	ad2antll 742	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
44		simprll 791	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑥 ∈ 𝒫 𝐽)
45	44	elpwid 4569	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑥 ⊆ 𝐽)
46	28, 45	sstrid 3924	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝐽)
47	43, 46	eqsstrd 3947	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ran 𝑔 ⊆ 𝐽)
48	47	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → ran 𝑔 ⊆ 𝐽)
49	41, 48	sstrid 3924	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ⊆ 𝐽)
50		fz1ssnn 13729	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
51		fof 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → 𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
52	51	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
53	52	fdmd 6750	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → dom 𝑔 = ℕ)
54	50, 53	sseqtrrid 3956	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (1...𝑛) ⊆ dom 𝑔)
55	54	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ dom 𝑔)
56		sseqin2 4123	. . . . . . . . . . . 12 ⊢ ((1...𝑛) ⊆ dom 𝑔 ↔ (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
57	55, 56	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛))
58		elfz1end 13728	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (1...𝑛))
59		ne0i 4290	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑛) → (1...𝑛) ≠ ∅)
60	59	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ (1...𝑛)) → (1...𝑛) ≠ ∅)
61	58, 60	sylan2b 606	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ≠ ∅)
62	57, 61	eqnetrd 3026	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
63		imadisj 6095	. . . . . . . . . . 11 ⊢ ((𝑔 “ (1...𝑛)) = ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) = ∅)
64	63	necon3bii 3011	. . . . . . . . . 10 ⊢ ((𝑔 “ (1...𝑛)) ≠ ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) ≠ ∅)
65	62, 64	sylibr 237	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ≠ ∅)
66		fzfid 14156	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
67	52	ffund 6744	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → Fun 𝑔)
68		fores 6836	. . . . . . . . . . 11 ⊢ ((Fun 𝑔 ∧ (1...𝑛) ⊆ dom 𝑔) → (𝑔 ↾ (1...𝑛)):(1...𝑛)⟶onto(𝑔 “ (1...𝑛)))
69	67, 55, 68	syl2an2r 698	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 ↾ (1...𝑛)):(1...𝑛)⟶onto(𝑔 “ (1...𝑛)))
70		fofi 9343	. . . . . . . . . 10 ⊢ (((1...𝑛) ∈ Fin ∧ (𝑔 ↾ (1...𝑛)):(1...𝑛)⟶onto(𝑔 “ (1...𝑛))) → (𝑔 “ (1...𝑛)) ∈ Fin)
71	66, 69, 70	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ Fin)
72		fiinopn 23274	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → (((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin) → ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽))
73	72	imp 412	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin)) → ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽)
74	40, 49, 65, 71, 73	syl13anc 1400	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽)
75	74	fmpttd 7147	. . . . . . 7 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))):ℕ⟶𝐽)
76		imassrn 6216	. . . . . . . . . . . . 13 ⊢ (𝑔 “ (1...𝑘)) ⊆ ran 𝑔
77	43	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
78	76, 77	sseqtrid 3955	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})
79		id 23	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛)
80	79	rgenw 3084	. . . . . . . . . . . . 13 ⊢ ∀𝑛 ∈ 𝑥 (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛)
81		eleq2w 2848	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑛 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑛))
82	81	ralrab 3655	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛 ↔ ∀𝑛 ∈ 𝑥 (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛))
83	80, 82	mpbir 234	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛
84		ssralv 3982	. . . . . . . . . . . 12 ⊢ ((𝑔 “ (1...𝑘)) ⊆ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → (∀𝑛 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛 → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛))
85	78, 83, 84	mpisyl 22	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛)
86		elintg 4920	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ∩ (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛))
87	86	ad3antlr 744	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ∩ (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛))
88	85, 87	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ∩ (𝑔 “ (1...𝑘)))
89		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))
90		oveq2 7459	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
91	90	imaeq2d 6070	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘)))
92	91	inteqd 4917	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ∩ (𝑔 “ (1...𝑛)) = ∩ (𝑔 “ (1...𝑘)))
93		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
94	74	ralrimiva 3158	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑛 ∈ ℕ ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽)
95	92	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ↔ ∩ (𝑔 “ (1...𝑘)) ∈ 𝐽))
96	95	rspccva 3579	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ 𝑘 ∈ ℕ) → ∩ (𝑔 “ (1...𝑘)) ∈ 𝐽)
97	94, 96	sylan 592	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔 “ (1...𝑘)) ∈ 𝐽)
98	89, 92, 93, 97	fvmptd3 7048	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) = ∩ (𝑔 “ (1...𝑘)))
99	88, 98	eleqtrrd 2867	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘))
100		fzssp1 13741	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ (1...(𝑘 + 1))
101		imass2 6073	. . . . . . . . . . . 12 ⊢ ((1...𝑘) ⊆ (1...(𝑘 + 1)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))))
102	100, 101	mp1i 14	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))))
103		intss 4934	. . . . . . . . . . 11 ⊢ ((𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))) → ∩ (𝑔 “ (1...(𝑘 + 1))) ⊆ ∩ (𝑔 “ (1...𝑘)))
104	102, 103	syl 18	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔 “ (1...(𝑘 + 1))) ⊆ ∩ (𝑔 “ (1...𝑘)))
105		oveq2 7459	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
106	105	imaeq2d 6070	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1))))
107	106	inteqd 4917	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → ∩ (𝑔 “ (1...𝑛)) = ∩ (𝑔 “ (1...(𝑘 + 1))))
108		peano2nn 12387	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
109	108	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
110	107	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → (∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ↔ ∩ (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽))
111	110	rspccva 3579	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ (𝑘 + 1) ∈ ℕ) → ∩ (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
112	94, 108, 111	syl2an 608	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)
113	89, 107, 109, 112	fvmptd3 7048	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) = ∩ (𝑔 “ (1...(𝑘 + 1))))
114	104, 113, 98	3sstr4d 3968	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘))
115	99, 114	jca 521	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)))
116	115	ralrimiva 3158	. . . . . . 7 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)))
117		simprlr 792	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
118		eleq2w 2848	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑦))
119		sseq2 3939	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ 𝑦))
120	119	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)))
121	120	rexbidv 3190	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)))
122	118, 121	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))))
123	122	rspccva 3579	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)))
124	117, 123	sylan 592	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)))
125		eleq2w 2848	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑤 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑤))
126	125	rexrab 3657	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))
127	43	rexeqdv 3324	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦))
128		fofn 6828	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → 𝑔 Fn ℕ)
129	128	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑔 Fn ℕ)
130		sseq1 3938	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑔‘𝑘) → (𝑤 ⊆ 𝑦 ↔ (𝑔‘𝑘) ⊆ 𝑦))
131	130	rexrn 7119	. . . . . . . . . . . . . . 15 ⊢ (𝑔 Fn ℕ → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦))
132	129, 131	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦))
133	127, 132	bitr3d 284	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦))
134	133	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦))
135		elfz1end 13728	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (1...𝑘))
136		fz1ssnn 13729	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑘) ⊆ ℕ
137	53	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → dom 𝑔 = ℕ)
138	136, 137	sseqtrrid 3956	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (1...𝑘) ⊆ dom 𝑔)
139		funfvima2 7268	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑔 ∧ (1...𝑘) ⊆ dom 𝑔) → (𝑘 ∈ (1...𝑘) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘))))
140	67, 138, 139	syl2an2r 698	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝑘 ∈ (1...𝑘) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘))))
141	140	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ (1...𝑘)) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)))
142	135, 141	sylan2b 606	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)))
143		intss1 4928	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)) → ∩ (𝑔 “ (1...𝑘)) ⊆ (𝑔‘𝑘))
144		sstr2 3920	. . . . . . . . . . . . . 14 ⊢ (∩ (𝑔 “ (1...𝑘)) ⊆ (𝑔‘𝑘) → ((𝑔‘𝑘) ⊆ 𝑦 → ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
145	142, 143, 144	3syl 19	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑔‘𝑘) ⊆ 𝑦 → ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
146	145	reximdva 3179	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
147	134, 146	sylbid 243	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 → ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
148	126, 147	biimtrrid 246	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦) → ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
149	124, 148	syld 48	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
150	98	sseq1d 3944	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
151	150	rexbidva 3188	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
152	151	adantr 486	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦))
153	149, 152	sylibrd 262	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
154	153	ralrimiva 3158	. . . . . . 7 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
155		nnex 12381	. . . . . . . . 9 ⊢ ℕ ∈ V
156	155	mptex 7260	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) ∈ V
157		feq1 6717	. . . . . . . . 9 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (𝑓:ℕ⟶𝐽 ↔ (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))):ℕ⟶𝐽))
158		fveq1 6915	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (𝑓‘𝑘) = ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘))
159	158	eleq2d 2850	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (𝐴 ∈ (𝑓‘𝑘) ↔ 𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)))
160		fveq1 6915	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (𝑓‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)))
161	160, 158	sseq12d 3946	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → ((𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘) ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)))
162	159, 161	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → ((𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ↔ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘))))
163	162	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ↔ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘))))
164	158	sseq1d 3944	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → ((𝑓‘𝑘) ⊆ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
165	164	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))
166	165	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → ((𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦) ↔ (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
167	166	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)))
168	157, 163, 167	3anbi123d 1465	. . . . . . . 8 ⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))) → ((𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)) ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦))))
169	156, 168	spcev 3564	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔 “ (1...𝑛)))‘𝑘) ⊆ 𝑦)) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))
170	75, 116, 154, 169	syl3anc 1399	. . . . . 6 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))
171	170	expr 462	. . . . 5 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → (𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))))
172	171	adantrrl 737	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → (𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))))
173	172	exlimdv 1968	. . 3 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → (∃𝑔 𝑔:ℕ⟶onto{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))))
174	39, 173	mpd 16	. 2 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))
175	2, 174	rexlimddv 3173	1 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4872  ∩ cint 4912   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  ωcom 7911   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010  Fincfn 9011  1c1 11240   + caddc 11242  ℕcn 12375  ...cfz 13679  Topctop 23266  1^stωc1stc 23810

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-top 23267  df-1stc 23812

This theorem is used by:  1stcelcls  23835