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Theorem 2ndcctbss 21537
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⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}
Assertion
Ref Expression
2ndcctbss ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (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.
StepHypRef Expression
1 simpr 477 . . 3 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → 𝐽 ∈ 2nd𝜔)
2 is2ndc 21528 . . 3 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
31, 2sylib 209 . 2 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4 vex 3352 . . . . . . 7 𝑐 ∈ V
54, 4xpex 7159 . . . . . 6 (𝑐 × 𝑐) ∈ V
6 3simpa 1178 . . . . . . . 8 ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) → (𝑢𝑐𝑣𝑐))
76ssopab2i 5163 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
8 2ndcctbss.2 . . . . . . 7 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}
9 df-xp 5282 . . . . . . 7 (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
107, 8, 93sstr4i 3803 . . . . . 6 𝑆 ⊆ (𝑐 × 𝑐)
11 ssdomg 8205 . . . . . 6 ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
125, 10, 11mp2 9 . . . . 5 𝑆 ≼ (𝑐 × 𝑐)
134xpdom1 8265 . . . . . . . . 9 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14 omex 8754 . . . . . . . . . 10 ω ∈ V
1514xpdom2 8261 . . . . . . . . 9 (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16 domtr 8212 . . . . . . . . 9 (((𝑐 × 𝑐) ≼ (ω × 𝑐) ∧ (ω × 𝑐) ≼ (ω × ω)) → (𝑐 × 𝑐) ≼ (ω × ω))
1713, 15, 16syl2anc 579 . . . . . . . 8 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × ω))
18 xpomen 9088 . . . . . . . 8 (ω × ω) ≈ ω
19 domentr 8218 . . . . . . . 8 (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
2017, 18, 19sylancl 580 . . . . . . 7 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
2120adantr 472 . . . . . 6 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
2221ad2antll 720 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23 domtr 8212 . . . . 5 ((𝑆 ≼ (𝑐 × 𝑐) ∧ (𝑐 × 𝑐) ≼ ω) → 𝑆 ≼ ω)
2412, 22, 23sylancr 581 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑆 ≼ ω)
258relopabi 5413 . . . . . . . . 9 Rel 𝑆
26 simpr 477 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → 𝑥𝑆)
27 1st2nd 7413 . . . . . . . . 9 ((Rel 𝑆𝑥𝑆) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2825, 26, 27sylancr 581 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2928, 26eqeltrrd 2844 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
30 df-br 4809 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
31 fvex 6387 . . . . . . . . . 10 (1st𝑥) ∈ V
32 fvex 6387 . . . . . . . . . 10 (2nd𝑥) ∈ V
33 simpl 474 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑢 = (1st𝑥))
3433eleq1d 2828 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑢𝑐 ↔ (1st𝑥) ∈ 𝑐))
35 simpr 477 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑣 = (2nd𝑥))
3635eleq1d 2828 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑣𝑐 ↔ (2nd𝑥) ∈ 𝑐))
37 sseq1 3785 . . . . . . . . . . . . 13 (𝑢 = (1st𝑥) → (𝑢𝑤 ↔ (1st𝑥) ⊆ 𝑤))
38 sseq2 3786 . . . . . . . . . . . . 13 (𝑣 = (2nd𝑥) → (𝑤𝑣𝑤 ⊆ (2nd𝑥)))
3937, 38bi2anan9 629 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑤𝑤𝑣) ↔ ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4039rexbidv 3198 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4134, 36, 403anbi123d 1560 . . . . . . . . . 10 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))))
4231, 32, 41, 8braba 5152 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4330, 42bitr3i 268 . . . . . . . 8 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆 ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4443simp3bi 1177 . . . . . . 7 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆 → ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
4529, 44syl 17 . . . . . 6 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
46 fvi 6443 . . . . . . . 8 (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
4746ad3antrrr 721 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ( I ‘𝐵) = 𝐵)
4847rexeqdv 3292 . . . . . 6 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → (∃𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)) ↔ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4945, 48mpbird 248 . . . . 5 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ∃𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
5049ralrimiva 3112 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
51 fvex 6387 . . . . 5 ( I ‘𝐵) ∈ V
52 sseq2 3786 . . . . . 6 (𝑤 = (𝑓𝑥) → ((1st𝑥) ⊆ 𝑤 ↔ (1st𝑥) ⊆ (𝑓𝑥)))
53 sseq1 3785 . . . . . 6 (𝑤 = (𝑓𝑥) → (𝑤 ⊆ (2nd𝑥) ↔ (𝑓𝑥) ⊆ (2nd𝑥)))
5452, 53anbi12d 624 . . . . 5 (𝑤 = (𝑓𝑥) → (((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)) ↔ ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5551, 54axcc4dom 9515 . . . 4 ((𝑆 ≼ ω ∧ ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5624, 50, 55syl2anc 579 . . 3 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5746ad2antrr 717 . . . . . . 7 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
5857feq3d 6209 . . . . . 6 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆𝐵))
5958anbi1d 623 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ↔ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))))
60 2ndctop 21529 . . . . . . . . . . . 12 (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top)
6160adantl 473 . . . . . . . . . . 11 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → 𝐽 ∈ Top)
6261ad2antrr 717 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐽 ∈ Top)
63 frn 6228 . . . . . . . . . . . 12 (𝑓:𝑆𝐵 → ran 𝑓𝐵)
6463ad2antrl 719 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐵)
65 bastg 21049 . . . . . . . . . . . . 13 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
6665ad3antrrr 721 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐵 ⊆ (topGen‘𝐵))
67 2ndcctbss.1 . . . . . . . . . . . 12 𝐽 = (topGen‘𝐵)
6866, 67syl6sseqr 3811 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐵𝐽)
6964, 68sstrd 3770 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐽)
70 simprrl 799 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑜𝐽)
71 simprr 789 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽)) → (topGen‘𝑐) = 𝐽)
7271ad2antlr 718 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → (topGen‘𝑐) = 𝐽)
7370, 72eleqtrrd 2846 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑜 ∈ (topGen‘𝑐))
74 simprrr 800 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑡𝑜)
75 tg2 21048 . . . . . . . . . . . . . 14 ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡𝑜) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
7673, 74, 75syl2anc 579 . . . . . . . . . . . . 13 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
77 bastg 21049 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
7877ad2antrl 719 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
7978ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑐 ⊆ (topGen‘𝑐))
8067eqeq2i 2776 . . . . . . . . . . . . . . . . . . . . 21 ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
8180biimpi 207 . . . . . . . . . . . . . . . . . . . 20 ((topGen‘𝑐) = 𝐽 → (topGen‘𝑐) = (topGen‘𝐵))
8281adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (topGen‘𝑐) = (topGen‘𝐵))
8382ad2antll 720 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
8483ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → (topGen‘𝑐) = (topGen‘𝐵))
8579, 84sseqtrd 3800 . . . . . . . . . . . . . . . 16 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑐 ⊆ (topGen‘𝐵))
86 simprl 787 . . . . . . . . . . . . . . . 16 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑑𝑐)
8785, 86sseldd 3761 . . . . . . . . . . . . . . 15 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑑 ∈ (topGen‘𝐵))
88 simprrl 799 . . . . . . . . . . . . . . 15 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑡𝑑)
89 tg2 21048 . . . . . . . . . . . . . . 15 ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡𝑑) → ∃𝑚𝐵 (𝑡𝑚𝑚𝑑))
9087, 88, 89syl2anc 579 . . . . . . . . . . . . . 14 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → ∃𝑚𝐵 (𝑡𝑚𝑚𝑑))
9165ad3antrrr 721 . . . . . . . . . . . . . . . . . . 19 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝐵 ⊆ (topGen‘𝐵))
9291ad2antrr 717 . . . . . . . . . . . . . . . . . 18 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝐵 ⊆ (topGen‘𝐵))
9372ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → (topGen‘𝑐) = 𝐽)
9493, 67syl6req 2815 . . . . . . . . . . . . . . . . . 18 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → (topGen‘𝐵) = (topGen‘𝑐))
9592, 94sseqtrd 3800 . . . . . . . . . . . . . . . . 17 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝐵 ⊆ (topGen‘𝑐))
96 simprl 787 . . . . . . . . . . . . . . . . 17 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑚𝐵)
9795, 96sseldd 3761 . . . . . . . . . . . . . . . 16 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑚 ∈ (topGen‘𝑐))
98 simprrl 799 . . . . . . . . . . . . . . . 16 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑡𝑚)
99 tg2 21048 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡𝑚) → ∃𝑛𝑐 (𝑡𝑛𝑛𝑚))
10097, 98, 99syl2anc 579 . . . . . . . . . . . . . . 15 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → ∃𝑛𝑐 (𝑡𝑛𝑛𝑚))
101 ffn 6222 . . . . . . . . . . . . . . . . . . . 20 (𝑓:𝑆𝐵𝑓 Fn 𝑆)
102101ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → 𝑓 Fn 𝑆)
103102ad2antlr 718 . . . . . . . . . . . . . . . . . 18 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑓 Fn 𝑆)
104103ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑓 Fn 𝑆)
105 simprl 787 . . . . . . . . . . . . . . . . . 18 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑐)
10686ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑑𝑐)
107 simplrl 795 . . . . . . . . . . . . . . . . . . 19 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝐵)
108 simprrr 800 . . . . . . . . . . . . . . . . . . 19 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑚)
109 simprr 789 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → 𝑚𝑑)
110109ad2antlr 718 . . . . . . . . . . . . . . . . . . 19 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝑑)
111 sseq2 3786 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑛𝑤𝑛𝑚))
112 sseq1 3785 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑤𝑑𝑚𝑑))
113111, 112anbi12d 624 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑚 → ((𝑛𝑤𝑤𝑑) ↔ (𝑛𝑚𝑚𝑑)))
114113rspcev 3460 . . . . . . . . . . . . . . . . . . 19 ((𝑚𝐵 ∧ (𝑛𝑚𝑚𝑑)) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
115107, 108, 110, 114syl12anc 865 . . . . . . . . . . . . . . . . . 18 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
116 df-br 4809 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
117 vex 3352 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
118 vex 3352 . . . . . . . . . . . . . . . . . . . 20 𝑑 ∈ V
119 simpl 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑢 = 𝑛)
120119eleq1d 2828 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑢𝑐𝑛𝑐))
121 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑣 = 𝑑)
122121eleq1d 2828 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑣𝑐𝑑𝑐))
123 sseq1 3785 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑛 → (𝑢𝑤𝑛𝑤))
124 sseq2 3786 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑑 → (𝑤𝑣𝑤𝑑))
125123, 124bi2anan9 629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑤𝑤𝑣) ↔ (𝑛𝑤𝑤𝑑)))
126125rexbidv 3198 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
127120, 122, 1263anbi123d 1560 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))))
128117, 118, 127, 8braba 5152 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
129116, 128bitr3i 268 . . . . . . . . . . . . . . . . . 18 (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
130105, 106, 115, 129syl3anbrc 1443 . . . . . . . . . . . . . . . . 17 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
131 fnfvelrn 6545 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
132104, 130, 131syl2anc 579 . . . . . . . . . . . . . . . 16 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
133 simprl 787 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑐)
134 simplll 791 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑑𝑐)
135 simplrl 795 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝐵)
136 simprrr 800 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑚)
137109ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝑑)
138135, 136, 137, 114syl12anc 865 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
139133, 134, 138, 129syl3anbrc 1443 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
140 fveq2 6374 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (1st𝑥) = (1st ‘⟨𝑛, 𝑑⟩))
141 fveq2 6374 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
142140, 141sseq12d 3793 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((1st𝑥) ⊆ (𝑓𝑥) ↔ (1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
143 fveq2 6374 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (2nd𝑥) = (2nd ‘⟨𝑛, 𝑑⟩))
144141, 143sseq12d 3793 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓𝑥) ⊆ (2nd𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)))
145142, 144anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑛, 𝑑⟩ → (((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) ↔ ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
146145rspcv 3456 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑛, 𝑑⟩ ∈ 𝑆 → (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
147139, 146syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
148117, 118op1st 7373 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑛, 𝑑⟩) = 𝑛
149148sseq1i 3788 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
150117, 118op2nd 7374 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑛, 𝑑⟩) = 𝑑
151150sseq2i 3789 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
152149, 151anbi12i 620 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)) ↔ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑))
153 simprl 787 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
154 simprl 787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → 𝑡𝑛)
155154ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡𝑛)
156153, 155sseldd 3761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩))
157 simprr 789 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
158 simplrr 796 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑑𝑜)
159158ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑑𝑜)
160157, 159sstrd 3770 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)
161156, 160jca 507 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
162161ex 401 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ((𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
163152, 162syl5bi 233 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
164147, 163syldc 48 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
165164exp4c 423 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
166165ad2antlr 718 . . . . . . . . . . . . . . . . . 18 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
167166adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
168167imp41 416 . . . . . . . . . . . . . . . 16 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
169 eleq2 2832 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡𝑏𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
170 sseq1 3785 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑏𝑜 ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
171169, 170anbi12d 624 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → ((𝑡𝑏𝑏𝑜) ↔ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
172171rspcev 3460 . . . . . . . . . . . . . . . 16 (((𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓 ∧ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
173132, 168, 172syl2anc 579 . . . . . . . . . . . . . . 15 (((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
174100, 173rexlimddv 3181 . . . . . . . . . . . . . 14 ((((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
17590, 174rexlimddv 3181 . . . . . . . . . . . . 13 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
17676, 175rexlimddv 3181 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
177176expr 448 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ((𝑜𝐽𝑡𝑜) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜)))
178177ralrimivv 3116 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ∀𝑜𝐽𝑡𝑜𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
179 basgen2 21072 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ∀𝑜𝐽𝑡𝑜𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜)) → (topGen‘ran 𝑓) = 𝐽)
18062, 69, 178, 179syl3anc 1490 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → (topGen‘ran 𝑓) = 𝐽)
181180, 62eqeltrd 2843 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → (topGen‘ran 𝑓) ∈ Top)
182 tgclb 21053 . . . . . . . 8 (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
183181, 182sylibr 225 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ∈ TopBases)
184 omelon 8757 . . . . . . . . . 10 ω ∈ On
18524adantr 472 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ≼ ω)
186 ondomen 9110 . . . . . . . . . 10 ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
187184, 185, 186sylancr 581 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ∈ dom card)
188101ad2antrl 719 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑓 Fn 𝑆)
189 dffn4 6303 . . . . . . . . . 10 (𝑓 Fn 𝑆𝑓:𝑆onto→ran 𝑓)
190188, 189sylib 209 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑓:𝑆onto→ran 𝑓)
191 fodomnum 9130 . . . . . . . . 9 (𝑆 ∈ dom card → (𝑓:𝑆onto→ran 𝑓 → ran 𝑓𝑆))
192187, 190, 191sylc 65 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝑆)
193 domtr 8212 . . . . . . . 8 ((ran 𝑓𝑆𝑆 ≼ ω) → ran 𝑓 ≼ ω)
194192, 185, 193syl2anc 579 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ≼ ω)
195180eqcomd 2770 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐽 = (topGen‘ran 𝑓))
196 breq1 4811 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
197 sseq1 3785 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏𝐵 ↔ ran 𝑓𝐵))
198 fveq2 6374 . . . . . . . . . 10 (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
199198eqeq2d 2774 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
200196, 197, 1993anbi123d 1560 . . . . . . . 8 (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))))
201200rspcev 3460 . . . . . . 7 ((ran 𝑓 ∈ TopBases ∧ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
202183, 194, 64, 195, 201syl13anc 1491 . . . . . 6 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
203202ex 401 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏))))
20459, 203sylbid 231 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ((𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏))))
205204exlimdv 2028 . . 3 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏))))
20656, 205mpd 15 . 2 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
2073, 206rexlimddv 3181 1 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3054  wrex 3055  Vcvv 3349  wss 3731  cop 4339   class class class wbr 4808  {copab 4870   I cid 5183   × cxp 5274  dom cdm 5276  ran crn 5277  Rel wrel 5281  Oncon0 5907   Fn wfn 6062  wf 6063  ontowfo 6065  cfv 6067  ωcom 7262  1st c1st 7363  2nd c2nd 7364  cen 8156  cdom 8157  cardccrd 9011  topGenctg 16365  Topctop 20976  TopBasesctb 21028  2nd𝜔c2ndc 21520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cc 9509
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-oi 8621  df-card 9015  df-acn 9018  df-topgen 16371  df-top 20977  df-bases 21029  df-2ndc 21522
This theorem is referenced by: (None)
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