Theorem 2ndcctbss 22514

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 ∈ 2ndω) → ∃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 484	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ 2ndω)
2		is2ndc 22505	. . 3 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
3	1, 2	sylib 217	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4		vex 3426	. . . . . . 7 ⊢ 𝑐 ∈ V
5	4, 4	xpex 7581	. . . . . 6 ⊢ (𝑐 × 𝑐) ∈ V
6		3simpa 1146	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐))
7	6	ssopab2i 5456	. . . . . . 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 5586	. . . . . . 7 ⊢ (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐)}
10	7, 8, 9	3sstr4i 3960	. . . . . 6 ⊢ 𝑆 ⊆ (𝑐 × 𝑐)
11		ssdomg 8741	. . . . . 6 ⊢ ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
12	5, 10, 11	mp2 9	. . . . 5 ⊢ 𝑆 ≼ (𝑐 × 𝑐)
13	4	xpdom1 8811	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14		omex 9331	. . . . . . . . . 10 ⊢ ω ∈ V
15	14	xpdom2 8807	. . . . . . . . 9 ⊢ (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16		domtr 8748	. . . . . . . . 9 ⊢ (((𝑐 × 𝑐) ≼ (ω × 𝑐) ∧ (ω × 𝑐) ≼ (ω × ω)) → (𝑐 × 𝑐) ≼ (ω × ω))
17	13, 15, 16	syl2anc 583	. . . . . . . 8 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × ω))
18		xpomen 9702	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
19		domentr 8754	. . . . . . . 8 ⊢ (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
20	17, 18, 19	sylancl 585	. . . . . . 7 ⊢ (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
21	20	adantr 480	. . . . . 6 ⊢ ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
22	21	ad2antll 725	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23		domtr 8748	. . . . 5 ⊢ ((𝑆 ≼ (𝑐 × 𝑐) ∧ (𝑐 × 𝑐) ≼ ω) → 𝑆 ≼ ω)
24	12, 22, 23	sylancr 586	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑆 ≼ ω)
25	8	relopabiv 5719	. . . . . . . . 9 ⊢ Rel 𝑆
26		simpr 484	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
27		1st2nd 7853	. . . . . . . . 9 ⊢ ((Rel 𝑆 ∧ 𝑥 ∈ 𝑆) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
28	25, 26, 27	sylancr 586	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
29	28, 26	eqeltrrd 2840	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝑆)
30		df-br 5071	. . . . . . . . 9 ⊢ ((1st ‘𝑥)𝑆(2nd ‘𝑥) ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝑆)
31		fvex 6769	. . . . . . . . . 10 ⊢ (1st ‘𝑥) ∈ V
32		fvex 6769	. . . . . . . . . 10 ⊢ (2nd ‘𝑥) ∈ V
33		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → 𝑢 = (1st ‘𝑥))
34	33	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → (𝑢 ∈ 𝑐 ↔ (1st ‘𝑥) ∈ 𝑐))
35		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → 𝑣 = (2nd ‘𝑥))
36	35	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → (𝑣 ∈ 𝑐 ↔ (2nd ‘𝑥) ∈ 𝑐))
37		sseq1 3942	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑥) → (𝑢 ⊆ 𝑤 ↔ (1st ‘𝑥) ⊆ 𝑤))
38		sseq2 3943	. . . . . . . . . . . . 13 ⊢ (𝑣 = (2nd ‘𝑥) → (𝑤 ⊆ 𝑣 ↔ 𝑤 ⊆ (2nd ‘𝑥)))
39	37, 38	bi2anan9 635	. . . . . . . . . . . 12 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → ((𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))))
40	39	rexbidv 3225	. . . . . . . . . . 11 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → (∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))))
41	34, 36, 40	3anbi123d 1434	. . . . . . . . . 10 ⊢ ((𝑢 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) → ((𝑢 ∈ 𝑐 ∧ 𝑣 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣)) ↔ ((1st ‘𝑥) ∈ 𝑐 ∧ (2nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)))))
42	31, 32, 41, 8	braba 5443	. . . . . . . . 9 ⊢ ((1st ‘𝑥)𝑆(2nd ‘𝑥) ↔ ((1st ‘𝑥) ∈ 𝑐 ∧ (2nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))))
43	30, 42	bitr3i 276	. . . . . . . 8 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝑆 ↔ ((1st ‘𝑥) ∈ 𝑐 ∧ (2nd ‘𝑥) ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))))
44	43	simp3bi 1145	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝑆 → ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)))
45	29, 44	syl 17	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)))
46		fvi 6826	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
47	46	ad3antrrr 726	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ( I ‘𝐵) = 𝐵)
48	47	rexeqdv 3340	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → (∃𝑤 ∈ ( I ‘𝐵)((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)) ↔ ∃𝑤 ∈ 𝐵 ((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))))
49	45, 48	mpbird 256	. . . . 5 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥 ∈ 𝑆) → ∃𝑤 ∈ ( I ‘𝐵)((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)))
50	49	ralrimiva 3107	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∀𝑥 ∈ 𝑆 ∃𝑤 ∈ ( I ‘𝐵)((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)))
51		fvex 6769	. . . . 5 ⊢ ( I ‘𝐵) ∈ V
52		sseq2 3943	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → ((1st ‘𝑥) ⊆ 𝑤 ↔ (1st ‘𝑥) ⊆ (𝑓‘𝑥)))
53		sseq1 3942	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ⊆ (2nd ‘𝑥) ↔ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))
54	52, 53	anbi12d 630	. . . . 5 ⊢ (𝑤 = (𝑓‘𝑥) → (((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥)) ↔ ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))))
55	51, 54	axcc4dom 10128	. . . 4 ⊢ ((𝑆 ≼ ω ∧ ∀𝑥 ∈ 𝑆 ∃𝑤 ∈ ( I ‘𝐵)((1st ‘𝑥) ⊆ 𝑤 ∧ 𝑤 ⊆ (2nd ‘𝑥))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))))
56	24, 50, 55	syl2anc 583	. . 3 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))))
57	46	ad2antrr 722	. . . . . . 7 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
58	57	feq3d 6571	. . . . . 6 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆⟶𝐵))
59	58	anbi1d 629	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ↔ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))))
60		2ndctop 22506	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ Top)
61	60	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ Top)
62	61	ad2antrr 722	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝐽 ∈ Top)
63		frn 6591	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶𝐵 → ran 𝑓 ⊆ 𝐵)
64	63	ad2antrl 724	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ran 𝑓 ⊆ 𝐵)
65		bastg 22024	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
66	65	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝐵 ⊆ (topGen‘𝐵))
67		2ndcctbss.1	. . . . . . . . . . . 12 ⊢ 𝐽 = (topGen‘𝐵)
68	66, 67	sseqtrrdi 3968	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝐵 ⊆ 𝐽)
69	64, 68	sstrd 3927	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ran 𝑓 ⊆ 𝐽)
70		simprrl 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → (topGen‘𝑐) = 𝐽)
73	70, 72	eleqtrrd 2842	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝑜 ∈ (topGen‘𝑐))
74		simprrr 778	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝑡 ∈ 𝑜)
75		tg2 22023	. . . . . . . . . . . . . 14 ⊢ ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑜) → ∃𝑑 ∈ 𝑐 (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))
76	73, 74, 75	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → ∃𝑑 ∈ 𝑐 (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))
77		bastg 22024	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
78	77	ad2antrl 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
79	78	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑐 ⊆ (topGen‘𝑐))
80	67	eqeq2i 2751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
81	80	biimpi 215	. . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
84	83	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → (topGen‘𝑐) = (topGen‘𝐵))
85	79, 84	sseqtrd 3957	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑐 ⊆ (topGen‘𝐵))
86		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑑 ∈ 𝑐)
87	85, 86	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑑 ∈ (topGen‘𝐵))
88		simprrl 777	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑡 ∈ 𝑑)
89		tg2 22023	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑑) → ∃𝑚 ∈ 𝐵 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))
90	87, 88, 89	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → 𝐵 ⊆ (topGen‘𝐵))
92	91	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → (topGen‘𝑐) = 𝐽)
94	93, 67	eqtr2di 2796	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 3957	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑚 ∈ 𝐵)
97	95, 96	sseldd 3918	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) → 𝑡 ∈ 𝑚)
99		tg2 22023	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡 ∈ 𝑚) → ∃𝑛 ∈ 𝑐 (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))
100	97, 98, 99	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 6584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑆⟶𝐵 → 𝑓 Fn 𝑆)
102	101	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜)) → 𝑓 Fn 𝑆)
103	102	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → 𝑓 Fn 𝑆)
104	103	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 3943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑚 → (𝑛 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑚))
112		sseq1 3942	. . . . . . . . . . . . . . . . . . . . 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 3552	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ 𝐵 ∧ (𝑛 ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑑)) → ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑))
115	107, 108, 110, 114	syl12anc 833	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 5071	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
117		vex 3426	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑛 ∈ V
118		vex 3426	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑑 ∈ V
119		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑢 = 𝑛)
120	119	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑢 ∈ 𝑐 ↔ 𝑛 ∈ 𝑐))
121		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → 𝑣 = 𝑑)
122	121	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (𝑣 ∈ 𝑐 ↔ 𝑑 ∈ 𝑐))
123		sseq1 3942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑛 → (𝑢 ⊆ 𝑤 ↔ 𝑛 ⊆ 𝑤))
124		sseq2 3943	. . . . . . . . . . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 = 𝑛 ∧ 𝑣 = 𝑑) → (∃𝑤 ∈ 𝐵 (𝑢 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑣) ↔ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
127	120, 122, 126	3anbi123d 1434	. . . . . . . . . . . . . . . . . . . 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 5443	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛𝑆𝑑 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
129	116, 128	bitr3i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛 ∈ 𝑐 ∧ 𝑑 ∈ 𝑐 ∧ ∃𝑤 ∈ 𝐵 (𝑛 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑑)))
130	105, 106, 115, 129	syl3anbrc 1341	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 6940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
132	104, 130, 131	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 1341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
140		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (1st ‘𝑥) = (1st ‘⟨𝑛, 𝑑⟩))
141		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓‘𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
142	140, 141	sseq12d 3950	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ↔ (1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
143		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑛, 𝑑⟩))
144	141, 143	sseq12d 3950	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓‘𝑥) ⊆ (2nd ‘𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)))
145	142, 144	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑛, 𝑑⟩ → (((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)) ↔ ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
146	145	rspcv 3547	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑛, 𝑑⟩ ∈ 𝑆 → (∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)) → ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨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 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)) → ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
148	117, 118	op1st 7812	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑛, 𝑑⟩) = 𝑛
149	148	sseq1i 3945	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
150	117, 118	op2nd 7813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑛, 𝑑⟩) = 𝑑
151	150	sseq2i 3946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
152	149, 151	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨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 3918	. . . . . . . . . . . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜)) ∧ (𝑚 ∈ 𝐵 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑑))) ∧ (𝑛 ∈ 𝑐 ∧ (𝑡 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑚))) → (((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
164	147, 163	syldc 48	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 432	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 425	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡 ∈ 𝑏 ↔ 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
170		sseq1 3942	. . . . . . . . . . . . . . . . . 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 3552	. . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 3219	. . . . . . . . . . . . . 14 ⊢ ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘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 3219	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) ∧ (𝑑 ∈ 𝑐 ∧ (𝑡 ∈ 𝑑 ∧ 𝑑 ⊆ 𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
176	76, 175	rexlimddv 3219	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) ∧ (𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
177	176	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ((𝑜 ∈ 𝐽 ∧ 𝑡 ∈ 𝑜) → ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜)))
178	177	ralrimivv 3113	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ∀𝑜 ∈ 𝐽 ∀𝑡 ∈ 𝑜 ∃𝑏 ∈ ran 𝑓(𝑡 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑜))
179		basgen2 22047	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → (topGen‘ran 𝑓) = 𝐽)
181	180, 62	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → (topGen‘ran 𝑓) ∈ Top)
182		tgclb 22028	. . . . . . . 8 ⊢ (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
183	181, 182	sylibr 233	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ran 𝑓 ∈ TopBases)
184		omelon 9334	. . . . . . . . . 10 ⊢ ω ∈ On
185	24	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝑆 ≼ ω)
186		ondomen 9724	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
187	184, 185, 186	sylancr 586	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝑆 ∈ dom card)
188	101	ad2antrl 724	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝑓 Fn 𝑆)
189		dffn4 6678	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑆 ↔ 𝑓:𝑆–onto→ran 𝑓)
190	188, 189	sylib 217	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝑓:𝑆–onto→ran 𝑓)
191		fodomnum 9744	. . . . . . . . 9 ⊢ (𝑆 ∈ dom card → (𝑓:𝑆–onto→ran 𝑓 → ran 𝑓 ≼ 𝑆))
192	187, 190, 191	sylc 65	. . . . . . . 8 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ran 𝑓 ≼ 𝑆)
193		domtr 8748	. . . . . . . 8 ⊢ ((ran 𝑓 ≼ 𝑆 ∧ 𝑆 ≼ ω) → ran 𝑓 ≼ ω)
194	192, 185, 193	syl2anc 583	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ran 𝑓 ≼ ω)
195	180	eqcomd 2744	. . . . . . 7 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → 𝐽 = (topGen‘ran 𝑓))
196		breq1 5073	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
197		sseq1 3942	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝑏 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
198		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
199	198	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
200	196, 197, 199	3anbi123d 1434	. . . . . . . 8 ⊢ (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓 ⊆ 𝐵 ∧ 𝐽 = (topGen‘ran 𝑓))))
201	200	rspcev 3552	. . . . . . 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 1370	. . . . . 6 ⊢ ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥)))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
203	202	ex 412	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶𝐵 ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
204	59, 203	sylbid 239	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
205	204	exlimdv 1937	. . 3 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥 ∈ 𝑆 ((1st ‘𝑥) ⊆ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ⊆ (2nd ‘𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏))))
206	56, 205	mpd 15	. 2 ⊢ (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))
207	3, 206	rexlimddv 3219	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏 ⊆ 𝐵 ∧ 𝐽 = (topGen‘𝑏)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070  {copab 5132   I cid 5479   × cxp 5578  dom cdm 5580  ran crn 5581  Rel wrel 5585  Oncon0 6251   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  ωcom 7687  1^st c1st 7802  2^nd c2nd 7803   ≈ cen 8688   ≼ cdom 8689  cardccrd 9624  topGenctg 17065  Topctop 21950  TopBasesctb 22003  2^ndωc2ndc 22497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-card 9628  df-acn 9631  df-topgen 17071  df-top 21951  df-bases 22004  df-2ndc 22499

This theorem is referenced by: (None)