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Theorem 2ndcctbss 21068

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	⊢ 𝑋 = ∪ 𝐵
2ndcctbss.2	⊢ 𝐽 = (topGen‘𝐵)
2ndcctbss.3	⊢ 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) 𝑋(𝑤,𝑣,𝑢,𝑏,𝑐)

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 476	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → 𝐽 ∈ 2^nd𝜔)
2		is2ndc 21059	. . 3 ⊢ (𝐽 ∈ 2^nd𝜔 ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
3	1, 2	sylib 207	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4		vex 3176	. . . . . . 7 ⊢ 𝑐 ∈ V
5	4, 4	xpex 6860	. . . . . 6 ⊢ (𝑐 × 𝑐) ∈ V
6		3simpa 1051	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐))
7	6	ssopab2i 4928	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐)}
8		2ndcctbss.3	. . . . . . 7 ⊢ 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣))}
9		df-xp 5044	. . . . . . 7 ⊢ (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐)}
10	7, 8, 9	3sstr4i 3607	. . . . . 6 ⊢ 𝑆 ⊆ (𝑐 × 𝑐)
11		ssdomg 7887	. . . . . 6 ⊢ ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
12	5, 10, 11	mp2 9	. . . . 5 ⊢ 𝑆 ≼ (𝑐 × 𝑐)
13	4	xpdom1 7944	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14		omex 8423	. . . . . . . . . 10 ⊢ ω ∈ V
15	14	xpdom2 7940	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16		domtr 7895	. . . . . . . . 9 ⊢ (((𝑐 × 𝑐) ≼ (ω × 𝑐) ∧ (ω × 𝑐) ≼ (ω × ω)) → (𝑐 × 𝑐) ≼ (ω × ω))
17	13, 15, 16	syl2anc 691	. . . . . . . 8 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × ω))
18		xpomen 8721	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
19		domentr 7901	. . . . . . . 8 ⊢ (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
20	17, 18, 19	sylancl 693	. . . . . . 7 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
21	20	adantr 480	. . . . . 6 ⊢ ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
22	21	ad2antll 761	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23		domtr 7895	. . . . 5 ⊢ ((𝑆 ≼ (𝑐 × 𝑐) ∧ (𝑐 × 𝑐) ≼ ω) → 𝑆 ≼ ω)
24	12, 22, 23	sylancr 694	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑆 ≼ ω)
25	8	relopabi 5167	. . . . . . . . 9 ⊢ Rel 𝑆
26		simpr 476	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
27		1st2nd 7105	. . . . . . . . 9 ⊢ ((Rel 𝑆 ∧ 𝑥 ∈ 𝑆) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
28	25, 26, 27	sylancr 694	. . . . . . . 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 2689	. . . . . . 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 4584	. . . . . . . . 9 ⊢ ((1^st ‘𝑥)𝑆(2^nd ‘𝑥) ↔ ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑆)
31		fvex 6113	. . . . . . . . . 10 ⊢ (1^st ‘𝑥) ∈ V
32		fvex 6113	. . . . . . . . . 10 ⊢ (2^nd ‘𝑥) ∈ V
33		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → 𝑢 = (1^st ‘𝑥))
34	33	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → (𝑢 ∈ 𝑐 ↔ (1^st ‘𝑥) ∈ 𝑐))
35		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → 𝑣 = (2^nd ‘𝑥))
36	35	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → (𝑣 ∈ 𝑐 ↔ (2^nd ‘𝑥) ∈ 𝑐))
37		sseq1 3589	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1^st ‘𝑥) → (𝑢 ⊆ 𝑤 ↔ (1^st ‘𝑥) ⊆ 𝑤))
38		sseq2 3590	. . . . . . . . . . . . 13 ⊢ (𝑣 = (2^nd ‘𝑥) → (𝑤 ⊆ 𝑣 ↔ 𝑤 ⊆ (2^nd ‘𝑥)))
39	37, 38	bi2anan9 913	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1^st ‘𝑥) ∧ 𝑣 = (2^nd ‘𝑥)) → ((𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ((1^st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2^nd ‘𝑥))))
40	39	rexbidv 3034	. . . . . . . . . . 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 1391	. . . . . . . . . 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 4917	. . . . . . . . 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 265	. . . . . . . 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 1071	. . . . . . 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 6165	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
47	46	ad3antrrr 762	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ( I ‘𝐵) = 𝐵)
48	47	rexeqdv 3122	. . . . . 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 246	. . . . 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 2949	. . . 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 6113	. . . . 5 ⊢ ( I ‘𝐵) ∈ V
52		sseq2 3590	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → ((1^st ‘𝑥) ⊆ 𝑤 ↔ (1^st ‘𝑥) ⊆ (𝑓‘𝑥)))
53		sseq1 3589	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ⊆ (2^nd ‘𝑥) ↔ (𝑓‘𝑥) ⊆ (2^nd ‘𝑥)))
54	52, 53	anbi12d 743	. . . . 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 9146	. . . 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 691	. . 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 758	. . . . . . 7 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
58	57	feq3d 5945	. . . . . 6 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆⟶𝐵))
59	58	anbi1d 737	. . . . 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 21060	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ 2^nd𝜔 → 𝐽 ∈ Top)
61	60	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) → 𝐽 ∈ Top)
62	61	ad2antrr 758	. . . . . . . . . 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 5966	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶𝐵 → ran 𝑓 ⊆ 𝐵)
64	63	ad2antrl 760	. . . . . . . . . . 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 20581	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
66	65	ad3antrrr 762	. . . . . . . . . . . 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.2	. . . . . . . . . . . 12 ⊢ 𝐽 = (topGen‘𝐵)
68	66, 67	syl6sseqr 3615	. . . . . . . . . . 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 3578	. . . . . . . . . 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 800	. . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽)) → (topGen‘𝑐) = 𝐽)
72	71	ad2antlr 759	. . . . . . . . . . . . . . 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 2691	. . . . . . . . . . . . . 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 801	. . . . . . . . . . . . . 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 20580	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑜) → ∃𝑑 ∈ 𝑐 (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))
76	73, 74, 75	syl2anc 691	. . . . . . . . . . . . 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 20581	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
78	77	ad2antrl 760	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
79	78	ad2antrr 758	. . . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
81	80	biimpi 205	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((topGen‘𝑐) = 𝐽 → (topGen‘𝑐) = (topGen‘𝐵))
82	81	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (topGen‘𝑐) = (topGen‘𝐵))
83	82	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
84	83	ad2antrr 758	. . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . 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 20580	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑑) → ∃𝑚 ∈ 𝐵 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))
90	87, 88, 89	syl2anc 691	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . . 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 2661	. . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . 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 20580	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑚) → ∃𝑛 ∈ 𝑐 (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))
100	97, 98, 99	syl2anc 691	. . . . . . . . . . . . . . 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 5958	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑆⟶𝐵 → 𝑓 Fn 𝑆)
102	101	ad2antrr 758	. . . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . 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 801	. . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → 𝑚 ⊆ 𝑑)
110	109	ad2antlr 759	. . . . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑚 → (𝑛 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑚))
112		sseq1 3589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑚 → (𝑤 ⊆ 𝑑 ↔ 𝑚 ⊆ 𝑑))
113	111, 112	anbi12d 743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑚 → ((𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑) ↔ (𝑛 ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑑)))
114	113	rspcev 3282	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑛 ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))
115	107, 108, 110, 114	syl12anc 1316	. . . . . . . . . . . . . . . . . 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 4584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
117		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
118		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑑 ∈ V
119		simpl 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑢 = 𝑛)
120	119	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑢 ∈ 𝑐 ↔ 𝑛 ∈ 𝑐))
121		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑣 = 𝑑)
122	121	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑣 ∈ 𝑐 ↔ 𝑑 ∈ 𝑐))
123		sseq1 3589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑛 → (𝑢 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑤))
124		sseq2 3590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = 𝑑 → (𝑤 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑑))
125	123, 124	bi2anan9 913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → ((𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
126	125	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
127	120, 122, 126	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . 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 4917	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
129	116, 128	bitr3i 265	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
130	105, 106, 115, 129	syl3anbrc 1239	. . . . . . . . . . . . . . . . 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 6264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
132	104, 130, 131	syl2anc 691	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ∈ 𝑐)
134		simplll 794	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑑 ∈ 𝑐)
135		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑚 ∈ 𝐵)
136		simprrr 801	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → 𝑛 ⊆ 𝑚)
137	109	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((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 1316	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((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 1239	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
140		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (1^st ‘𝑥) = (1^st ‘⟨𝑛, 𝑑⟩))
141		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓‘𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
142	140, 141	sseq12d 3597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((1^st ‘𝑥) ⊆ (𝑓‘𝑥) ↔ (1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
143		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (2^nd ‘𝑥) = (2^nd ‘⟨𝑛, 𝑑⟩))
144	141, 143	sseq12d 3597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓‘𝑥) ⊆ (2^nd ‘𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩)))
145	142, 144	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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 3278	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨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	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((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 7067	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1^st ‘⟨𝑛, 𝑑⟩) = 𝑛
149	148	sseq1i 3592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
150	117, 118	op2nd 7068	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2^nd ‘⟨𝑛, 𝑑⟩) = 𝑑
151	150	sseq2i 3593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
152	149, 151	anbi12i 729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1^st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2^nd ‘⟨𝑛, 𝑑⟩)) ↔ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑))
153		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((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 790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚)) → 𝑡 ∈ 𝑛)
155	154	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((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 3569	. . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ (𝑓‘⟨𝑛, 𝑑⟩))
157		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((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 797	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑑 ⊆ 𝑜)
159	158	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((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 3578	. . . . . . . . . . . . . . . . . . . . . . . . 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⟩) ⊆ 𝑜)
161	156, 160	jca 553	. . . . . . . . . . . . . . . . . . . . . . . 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⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
162	161	ex 449	. . . . . . . . . . . . . . . . . . . . . . 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)))
163	152, 162	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((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	syld 46	. . . . . . . . . . . . . . . . . . . . 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)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
165	164	com12 32	. . . . . . . . . . . . . . . . . . . 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)))
166	165	exp4c 634	. . . . . . . . . . . . . . . . . . 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)))))
167	166	ad2antlr 759	. . . . . . . . . . . . . . . . . 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)))))
168	167	adantl 481	. . . . . . . . . . . . . . . . 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)))))
169	168	imp41 617	. . . . . . . . . . . . . . . 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))
170		eleq2 2677	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡 ∈ 𝑏 ↔ 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
171		sseq1 3589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑏 ⊆ 𝑜 ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
172	170, 171	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → ((𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜) ↔ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
173	172	rspcev 3282	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓 ∧ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
174	132, 169, 173	syl2anc 691	. . . . . . . . . . . . . . 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))
175	100, 174	rexlimddv 3017	. . . . . . . . . . . . . 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))
176	90, 175	rexlimddv 3017	. . . . . . . . . . . . 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))
177	76, 176	rexlimddv 3017	. . . . . . . . . . . 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))
178	177	expr 641	. . . . . . . . . . 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)))
179	178	ralrimivv 2953	. . . . . . . . . 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))
180		basgen2 20604	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ran 𝑓 ⊆ 𝐽 ∧ ∀𝑜 ∈ 𝐽 ∀𝑡 ∈ 𝑜 ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜)) → (topGen‘ran 𝑓) = 𝐽)
181	62, 69, 179, 180	syl3anc 1318	. . . . . . . . 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)
182	181, 62	eqeltrd 2688	. . . . . . . 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)
183		tgclb 20585	. . . . . . . 8 ⊢ (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
184	182, 183	sylibr 223	. . . . . . 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)
185		omelon 8426	. . . . . . . . . 10 ⊢ ω ∈ On
186	24	adantr 480	. . . . . . . . . 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)
187		ondomen 8743	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
188	185, 186, 187	sylancr 694	. . . . . . . . 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)
189	101	ad2antrl 760	. . . . . . . . . 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)
190		dffn4 6034	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑆 ↔ 𝑓:𝑆–onto→ran 𝑓)
191	189, 190	sylib 207	. . . . . . . . 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)
192		fodomnum 8763	. . . . . . . . 9 ⊢ (𝑆 ∈ dom card → (𝑓:𝑆–onto→ran 𝑓 → ran 𝑓 ≼ 𝑆))
193	188, 191, 192	sylc 63	. . . . . . . 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)
194		domtr 7895	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ 𝑆 ∧ 𝑆 ≼ ω) → ran 𝑓 ≼ ω)
195	193, 186, 194	syl2anc 691	. . . . . . 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)
196	181	eqcomd 2616	. . . . . . 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))
197		breq1 4586	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
198		sseq1 3589	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
199		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
200	199	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
201	197, 198, 200	3anbi123d 1391	. . . . . . . 8 ⊢ (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓 ⊆ 𝐵 ∧ 𝐽 = (topGen‘ran 𝑓))))
202	201	rspcev 3282	. . . . . . 7 ⊢ ((ran 𝑓 ∈ TopBases ∧ (ran 𝑓 ≼ ω ∧ ran 𝑓 ⊆ 𝐵 ∧ 𝐽 = (topGen‘ran 𝑓))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
203	184, 195, 64, 196, 202	syl13anc 1320	. . . . . 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)))
204	203	ex 449	. . . . 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))))
205	59, 204	sylbid 229	. . . 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))))
206	205	exlimdv 1848	. . 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))))
207	56, 206	mpd 15	. 2 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2^nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
208	3, 207	rexlimddv 3017	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 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ⟨cop 4131 ∪ cuni 4372 class class class wbr 4583 {copab 4642 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 Rel wrel 5043 Oncon0 5640 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 ωcom 6957 1^st c1st 7057 2^nd c2nd 7058 ≈ cen 7838 ≼ cdom 7839 cardccrd 8644 topGenctg 15921 Topctop 20517 TopBasesctb 20520 2^nd𝜔c2ndc 21051

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-card 8648 df-acn 8651 df-topgen 15927 df-top 20521 df-bases 20522 df-2ndc 21053

This theorem is referenced by: (None)

W3C validator