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Theorem 2ndcctbss 21747

Description: If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

**Hypotheses**
Ref	Expression
2ndcctbss.1	⊢ 𝐽 = (topGen‘𝐵)
2ndcctbss.2	⊢ 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))}

**Assertion**
Ref	Expression
2ndcctbss	⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))

Distinct variable groups: 𝑏,𝑐,𝑢,𝑣,𝑤,𝐵 𝐽,𝑏,𝑐 Allowed substitution hints: 𝑆(𝑤,𝑣,𝑢,𝑏,𝑐) 𝐽(𝑤,𝑣,𝑢)

Proof of Theorem 2ndcctbss Dummy variables 𝑑 𝑓 𝑚 𝑛 𝑜 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → 𝐽 ∈ 2^nd𝜔)
2		is2ndc 21738	. . 3 ⊢ (𝐽 ∈ 2^nd𝜔 ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
3	1, 2	sylib 219	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4		vex 3439	. . . . . . 7 ⊢ 𝑐 ∈ V
5	4, 4	xpex 7336	. . . . . 6 ⊢ (𝑐 × 𝑐) ∈ V
6		3simpa 1141	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐))
7	6	ssopab2i 5328	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐)}
8		2ndcctbss.2	. . . . . . 7 ⊢ 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))}
9		df-xp 5452	. . . . . . 7 ⊢ (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐)}
10	7, 8, 9	3sstr4i 3933	. . . . . 6 ⊢ 𝑆 ⊆ (𝑐 × 𝑐)
11		ssdomg 8406	. . . . . 6 ⊢ ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
12	5, 10, 11	mp2 9	. . . . 5 ⊢ 𝑆 ≼ (𝑐 × 𝑐)
13	4	xpdom1 8466	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14		omex 8955	. . . . . . . . . 10 ⊢ ω ∈ V
15	14	xpdom2 8462	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16		domtr 8413	. . . . . . . . 9 ⊢ (((𝑐 × 𝑐) ≼ (ω × 𝑐) ∧ (ω × 𝑐) ≼ (ω × ω)) → (𝑐 × 𝑐) ≼ (ω × ω))
17	13, 15, 16	syl2anc 584	. . . . . . . 8 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × ω))
18		xpomen 9290	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
19		domentr 8419	. . . . . . . 8 ⊢ (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
20	17, 18, 19	sylancl 586	. . . . . . 7 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
21	20	adantr 481	. . . . . 6 ⊢ ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
22	21	ad2antll 725	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23		domtr 8413	. . . . 5 ⊢ ((𝑆 ≼ (𝑐 × 𝑐) ∧ (𝑐 × 𝑐) ≼ ω) → 𝑆 ≼ ω)
24	12, 22, 23	sylancr 587	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑆 ≼ ω)
25	8	relopabi 5583	. . . . . . . . 9 ⊢ Rel 𝑆
26		simpr 485	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
27		1st2nd 7597	. . . . . . . . 9 ⊢ ((Rel 𝑆 ∧ 𝑥 ∈ 𝑆) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
28	25, 26, 27	sylancr 587	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
29	28, 26	eqeltrrd 2883	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑆)
30		df-br 4965	. . . . . . . . 9 ⊢ ((1^st ‘𝑥)𝑆(2^nd ‘𝑥) ↔ ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑆)
31		fvex 6554	. . . . . . . . . 10 ⊢ (1^st ‘𝑥) ∈ V
32		fvex 6554	. . . . . . . . . 10 ⊢ (2^nd ‘𝑥) ∈ V
33		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → 𝑢 = (1^st ‘𝑥))
34	33	eleq1d 2866	. . . . . . . . . . 11 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → (𝑢 ∈ 𝑐 ↔ (1^st ‘𝑥) ∈ 𝑐))
35		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → 𝑣 = (2^nd ‘𝑥))
36	35	eleq1d 2866	. . . . . . . . . . 11 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → (𝑣 ∈ 𝑐 ↔ (2^nd ‘𝑥) ∈ 𝑐))
37		sseq1 3915	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1^st ‘𝑥) → (𝑢 ⊆ 𝑤 ↔ (1^st ‘𝑥) ⊆ 𝑤))
38		sseq2 3916	. . . . . . . . . . . . 13 ⊢ (𝑣 = (2^nd ‘𝑥) → (𝑤 ⊆ 𝑣 ↔ 𝑤 ⊆ (2^nd ‘𝑥)))
39	37, 38	bi2anan9 635	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → ((𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
40	39	rexbidv 3259	. . . . . . . . . . 11 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → (∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
41	34, 36, 40	3anbi123d 1428	. . . . . . . . . 10 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) ↔ ((1^st ‘𝑥) ∈ 𝑐 ∧ (2^nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)))))
42	31, 32, 41, 8	braba 5317	. . . . . . . . 9 ⊢ ((1^st ‘𝑥)𝑆(2^nd ‘𝑥) ↔ ((1^st ‘𝑥) ∈ 𝑐 ∧ (2^nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
43	30, 42	bitr3i 278	. . . . . . . 8 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑆 ↔ ((1^st ‘𝑥) ∈ 𝑐 ∧ (2^nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
44	43	simp3bi 1140	. . . . . . 7 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑆 → ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)))
45	29, 44	syl 17	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)))
46		fvi 6610	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
47	46	ad3antrrr 726	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ( I ‘𝐵) = 𝐵)
48	47	rexeqdv 3375	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → (∃𝑤 ∈ ( I ‘𝐵)((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)) ↔ ∃𝑤 ∈ 𝐵 ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
49	45, 48	mpbird 258	. . . . 5 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ∃𝑤 ∈ ( I ‘𝐵)((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)))
50	49	ralrimiva 3148	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∀𝑥 ∈ 𝑆 ∃𝑤 ∈ ( I ‘𝐵)((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)))
51		fvex 6554	. . . . 5 ⊢ ( I ‘𝐵) ∈ V
52		sseq2 3916	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → ((1^st ‘𝑥) ⊆ 𝑤 ↔ (1^st ‘𝑥) ⊆ (𝑓‘𝑥)))
53		sseq1 3915	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ⊆ (2^nd ‘𝑥) ↔ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))
54	52, 53	anbi12d 630	. . . . 5 ⊢ (𝑤 = (𝑓‘𝑥) → (((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥)) ↔ ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))))
55	51, 54	axcc4dom 9712	. . . 4 ⊢ ((𝑆 ≼ ω ∧ ∀𝑥 ∈ 𝑆 ∃𝑤 ∈ ( I ‘𝐵)((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))))
56	24, 50, 55	syl2anc 584	. . 3 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))))
57	46	ad2antrr 722	. . . . . . 7 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
58	57	feq3d 6372	. . . . . 6 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆⟶𝐵))
59	58	anbi1d 629	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ↔ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))))
60		2ndctop 21739	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ 2^nd𝜔 → 𝐽 ∈ Top)
61	60	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → 𝐽 ∈ Top)
62	61	ad2antrr 722	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝐽 ∈ Top)
63		frn 6391	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶𝐵 → ran 𝑓 ⊆ 𝐵)
64	63	ad2antrl 724	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ran 𝑓 ⊆ 𝐵)
65		bastg 21258	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
66	65	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝐵 ⊆ (topGen‘𝐵))
67		2ndcctbss.1	. . . . . . . . . . . 12 ⊢ 𝐽 = (topGen‘𝐵)
68	66, 67	syl6sseqr 3941	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝐵 ⊆ 𝐽)
69	64, 68	sstrd 3901	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ran 𝑓 ⊆ 𝐽)
70		simprrl 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝑜 ∈ 𝐽)
71		simprr 769	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽)) → (topGen‘𝑐) = 𝐽)
72	71	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → (topGen‘𝑐) = 𝐽)
73	70, 72	eleqtrrd 2885	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝑜 ∈ (topGen‘𝑐))
74		simprrr 778	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝑡 ∈ 𝑜)
75		tg2 21257	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑜) → ∃𝑑 ∈ 𝑐 (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))
76	73, 74, 75	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → ∃𝑑 ∈ 𝑐 (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))
77		bastg 21258	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
78	77	ad2antrl 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
79	78	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑐 ⊆ (topGen‘𝑐))
80	67	eqeq2i 2806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
81	80	biimpi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘𝑐) = 𝐽 → (topGen‘𝑐) = (topGen‘𝐵))
82	81	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (topGen‘𝑐) = (topGen‘𝐵))
83	82	ad2antll 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
84	83	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → (topGen‘𝑐) = (topGen‘𝐵))
85	79, 84	sseqtrd 3930	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑐 ⊆ (topGen‘𝐵))
86		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑑 ∈ 𝑐)
87	85, 86	sseldd 3892	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑑 ∈ (topGen‘𝐵))
88		simprrl 777	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑡 ∈ 𝑑)
89		tg2 21257	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑑) → ∃𝑚 ∈ 𝐵 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))
90	87, 88, 89	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → ∃𝑚 ∈ 𝐵 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))
91	65	ad3antrrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝐵 ⊆ (topGen‘𝐵))
92	91	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝐵 ⊆ (topGen‘𝐵))
93	72	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → (topGen‘𝑐) = 𝐽)
94	93, 67	syl6req 2847	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → (topGen‘𝐵) = (topGen‘𝑐))
95	92, 94	sseqtrd 3930	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝐵 ⊆ (topGen‘𝑐))
96		simprl 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑚 ∈ 𝐵)
97	95, 96	sseldd 3892	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑚 ∈ (topGen‘𝑐))
98		simprrl 777	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑡 ∈ 𝑚)
99		tg2 21257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑚) → ∃𝑛 ∈ 𝑐 (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))
100	97, 98, 99	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → ∃𝑛 ∈ 𝑐 (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))
101		ffn 6385	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑆⟶𝐵 → 𝑓 Fn 𝑆)
102	101	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜)) → 𝑓 Fn 𝑆)
103	102	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑓 Fn 𝑆)
104	103	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑓 Fn 𝑆)
105		simprl 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ∈ 𝑐)
106	86	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑑 ∈ 𝑐)
107		simplrl 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑚 ∈ 𝐵)
108		simprrr 778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ⊆ 𝑚)
109		simprr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → 𝑚 ⊆ 𝑑)
110	109	ad2antlr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑚 ⊆ 𝑑)
111		sseq2 3916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑚 → (𝑛 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑚))
112		sseq1 3915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑚 → (𝑤 ⊆ 𝑑 ↔ 𝑚 ⊆ 𝑑))
113	111, 112	anbi12d 630	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑚 → ((𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑) ↔ (𝑛 ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑑)))
114	113	rspcev 3557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑛 ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))
115	107, 108, 110, 114	syl12anc 833	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))
116		df-br 4965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
117		vex 3439	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
118		vex 3439	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑑 ∈ V
119		simpl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑢 = 𝑛)
120	119	eleq1d 2866	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑢 ∈ 𝑐 ↔ 𝑛 ∈ 𝑐))
121		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑣 = 𝑑)
122	121	eleq1d 2866	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑣 ∈ 𝑐 ↔ 𝑑 ∈ 𝑐))
123		sseq1 3915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑛 → (𝑢 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑤))
124		sseq2 3916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = 𝑑 → (𝑤 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑑))
125	123, 124	bi2anan9 635	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → ((𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
126	125	rexbidv 3259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
127	120, 122, 126	3anbi123d 1428	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))))
128	117, 118, 127, 8	braba 5317	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
129	116, 128	bitr3i 278	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
130	105, 106, 115, 129	syl3anbrc 1336	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
131		fnfvelrn 6716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
132	104, 130, 131	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
133		simprl 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ∈ 𝑐)
134		simplll 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑑 ∈ 𝑐)
135		simplrl 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑚 ∈ 𝐵)
136		simprrr 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ⊆ 𝑚)
137	109	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑚 ⊆ 𝑑)
138	135, 136, 137, 114	syl12anc 833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))
139	133, 134, 138, 129	syl3anbrc 1336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
140		fveq2 6541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (1^st ‘𝑥) = (1^st ‘⟨𝑛, 𝑑⟩))
141		fveq2 6541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓‘𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
142	140, 141	sseq12d 3923	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ↔ (1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
143		fveq2 6541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (2^nd ‘𝑥) = (2^nd ‘⟨𝑛, 𝑑⟩))
144	141, 143	sseq12d 3923	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓‘𝑥) ⊆ (2^nd ‘𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩)))
145	142, 144	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)) ↔ ((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩))))
146	145	rspcv 3553	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑛, 𝑑⟩ ∈ 𝑆 → (∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)) → ((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩))))
147	139, 146	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)) → ((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩))))
148	117, 118	op1st 7556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1^st ‘⟨𝑛, 𝑑⟩) = 𝑛
149	148	sseq1i 3918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
150	117, 118	op2nd 7557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2^nd ‘⟨𝑛, 𝑑⟩) = 𝑑
151	150	sseq2i 3919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
152	149, 151	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩)) ↔ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑))
153		simprl 767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
154		simprl 767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚)) → 𝑡 ∈ 𝑛)
155	154	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡 ∈ 𝑛)
156	153, 155	sseldd 3892	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩))
157		simprr 769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
158		simplrr 774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑑 ⊆ 𝑜)
159	158	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑑 ⊆ 𝑜)
160	157, 159	sstrd 3901	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)
161	156, 160	jca 512	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
162	161	ex 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ((𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
163	152, 162	syl5bi 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
164	147, 163	syldc 48	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)) → ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
165	164	exp4c 433	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)) → ((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) → ((𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ((𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
166	165	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜)) → ((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) → ((𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ((𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
167	166	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → ((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) → ((𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ((𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
168	167	imp41 426	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
169		eleq2 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡 ∈ 𝑏 ↔ 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
170		sseq1 3915	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑏 ⊆ 𝑜 ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
171	169, 170	anbi12d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → ((𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜) ↔ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
172	171	rspcev 3557	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓 ∧ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
173	132, 168, 172	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
174	100, 173	rexlimddv 3253	. . . . . . . . . . . . . 14 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
175	90, 174	rexlimddv 3253	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
176	76, 175	rexlimddv 3253	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
177	176	expr 457	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ((𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜)))
178	177	ralrimivv 3156	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ∀𝑜 ∈ 𝐽 ∀𝑡 ∈ 𝑜 ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
179		basgen2 21281	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ran 𝑓 ⊆ 𝐽 ∧ ∀𝑜 ∈ 𝐽 ∀𝑡 ∈ 𝑜 ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜)) → (topGen‘ran 𝑓) = 𝐽)
180	62, 69, 178, 179	syl3anc 1364	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → (topGen‘ran 𝑓) = 𝐽)
181	180, 62	eqeltrd 2882	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → (topGen‘ran 𝑓) ∈ Top)
182		tgclb 21262	. . . . . . . 8 ⊢ (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
183	181, 182	sylibr 235	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ran 𝑓 ∈ TopBases)
184		omelon 8958	. . . . . . . . . 10 ⊢ ω ∈ On
185	24	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝑆 ≼ ω)
186		ondomen 9312	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
187	184, 185, 186	sylancr 587	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝑆 ∈ dom card)
188	101	ad2antrl 724	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝑓 Fn 𝑆)
189		dffn4 6467	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑆 ↔ 𝑓:𝑆–onto→ran 𝑓)
190	188, 189	sylib 219	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝑓:𝑆–onto→ran 𝑓)
191		fodomnum 9332	. . . . . . . . 9 ⊢ (𝑆 ∈ dom card → (𝑓:𝑆–onto→ran 𝑓 → ran 𝑓 ≼ 𝑆))
192	187, 190, 191	sylc 65	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ran 𝑓 ≼ 𝑆)
193		domtr 8413	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ 𝑆 ∧ 𝑆 ≼ ω) → ran 𝑓 ≼ ω)
194	192, 185, 193	syl2anc 584	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ran 𝑓 ≼ ω)
195	180	eqcomd 2800	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → 𝐽 = (topGen‘ran 𝑓))
196		breq1 4967	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
197		sseq1 3915	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
198		fveq2 6541	. . . . . . . . . 10 ⊢ (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
199	198	eqeq2d 2804	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
200	196, 197, 199	3anbi123d 1428	. . . . . . . 8 ⊢ (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓 ⊆ 𝐵 ∧ 𝐽 = (topGen‘ran 𝑓))))
201	200	rspcev 3557	. . . . . . 7 ⊢ ((ran 𝑓 ∈ TopBases ∧ (ran 𝑓 ≼ ω ∧ ran 𝑓 ⊆ 𝐵 ∧ 𝐽 = (topGen‘ran 𝑓))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
202	183, 194, 64, 195, 201	syl13anc 1365	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
203	202	ex 413	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
204	59, 203	sylbid 241	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
205	204	exlimdv 1912	. . 3 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
206	56, 205	mpd 15	. 2 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
207	3, 206	rexlimddv 3253	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∃wex 1762 ∈ wcel 2080 ∀wral 3104 ∃wrex 3105 Vcvv 3436 ⊆ wss 3861 ⟨cop 4480 class class class wbr 4964 {copab 5026 I cid 5350 × cxp 5444 dom cdm 5446 ran crn 5447 Rel wrel 5451 Oncon0 6069 Fn wfn 6223 ⟶wf 6224 –onto→wfo 6226 ‘cfv 6228 ωcom 7439 1^st c1st 7546 2^nd c2nd 7547 ≈ cen 8357 ≼ cdom 8358 cardccrd 9213 topGenctg 16540 Topctop 21185 TopBasesctb 21237 2^nd𝜔c2ndc 21730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cc 9706

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-oi 8823 df-card 9217 df-acn 9220 df-topgen 16546 df-top 21186 df-bases 21238 df-2ndc 21732

This theorem is referenced by: (None)

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