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Theorem 2ndcctbss 23479
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 484 . . 3 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ 2ndω)
2 is2ndc 23470 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
31, 2sylib 218 . 2 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4 vex 3482 . . . . . . 7 𝑐 ∈ V
54, 4xpex 7772 . . . . . 6 (𝑐 × 𝑐) ∈ V
6 3simpa 1147 . . . . . . . 8 ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) → (𝑢𝑐𝑣𝑐))
76ssopab2i 5560 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
8 2ndcctbss.2 . . . . . . 7 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}
9 df-xp 5695 . . . . . . 7 (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
107, 8, 93sstr4i 4039 . . . . . 6 𝑆 ⊆ (𝑐 × 𝑐)
11 ssdomg 9039 . . . . . 6 ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
125, 10, 11mp2 9 . . . . 5 𝑆 ≼ (𝑐 × 𝑐)
134xpdom1 9110 . . . . . . . . 9 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14 omex 9681 . . . . . . . . . 10 ω ∈ V
1514xpdom2 9106 . . . . . . . . 9 (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16 domtr 9046 . . . . . . . . 9 (((𝑐 × 𝑐) ≼ (ω × 𝑐) ∧ (ω × 𝑐) ≼ (ω × ω)) → (𝑐 × 𝑐) ≼ (ω × ω))
1713, 15, 16syl2anc 584 . . . . . . . 8 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × ω))
18 xpomen 10053 . . . . . . . 8 (ω × ω) ≈ ω
19 domentr 9052 . . . . . . . 8 (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
2017, 18, 19sylancl 586 . . . . . . 7 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
2120adantr 480 . . . . . 6 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
2221ad2antll 729 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23 domtr 9046 . . . . 5 ((𝑆 ≼ (𝑐 × 𝑐) ∧ (𝑐 × 𝑐) ≼ ω) → 𝑆 ≼ ω)
2412, 22, 23sylancr 587 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑆 ≼ ω)
258relopabiv 5833 . . . . . . . . 9 Rel 𝑆
26 simpr 484 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → 𝑥𝑆)
27 1st2nd 8063 . . . . . . . . 9 ((Rel 𝑆𝑥𝑆) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2825, 26, 27sylancr 587 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2928, 26eqeltrrd 2840 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
30 df-br 5149 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
31 fvex 6920 . . . . . . . . . 10 (1st𝑥) ∈ V
32 fvex 6920 . . . . . . . . . 10 (2nd𝑥) ∈ V
33 simpl 482 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑢 = (1st𝑥))
3433eleq1d 2824 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑢𝑐 ↔ (1st𝑥) ∈ 𝑐))
35 simpr 484 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑣 = (2nd𝑥))
3635eleq1d 2824 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑣𝑐 ↔ (2nd𝑥) ∈ 𝑐))
37 sseq1 4021 . . . . . . . . . . . . 13 (𝑢 = (1st𝑥) → (𝑢𝑤 ↔ (1st𝑥) ⊆ 𝑤))
38 sseq2 4022 . . . . . . . . . . . . 13 (𝑣 = (2nd𝑥) → (𝑤𝑣𝑤 ⊆ (2nd𝑥)))
3937, 38bi2anan9 638 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑤𝑤𝑣) ↔ ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4039rexbidv 3177 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4134, 36, 403anbi123d 1435 . . . . . . . . . 10 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))))
4231, 32, 41, 8braba 5547 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4330, 42bitr3i 277 . . . . . . . 8 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆 ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4443simp3bi 1146 . . . . . . 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 6985 . . . . . . 7 (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
4746ad3antrrr 730 . . . . . 6 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ( I ‘𝐵) = 𝐵)
4845, 47rexeqtrrdv 3329 . . . . 5 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ∃𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
4948ralrimiva 3144 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
50 fvex 6920 . . . . 5 ( I ‘𝐵) ∈ V
51 sseq2 4022 . . . . . 6 (𝑤 = (𝑓𝑥) → ((1st𝑥) ⊆ 𝑤 ↔ (1st𝑥) ⊆ (𝑓𝑥)))
52 sseq1 4021 . . . . . 6 (𝑤 = (𝑓𝑥) → (𝑤 ⊆ (2nd𝑥) ↔ (𝑓𝑥) ⊆ (2nd𝑥)))
5351, 52anbi12d 632 . . . . 5 (𝑤 = (𝑓𝑥) → (((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)) ↔ ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5450, 53axcc4dom 10479 . . . 4 ((𝑆 ≼ ω ∧ ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5524, 49, 54syl2anc 584 . . 3 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5646ad2antrr 726 . . . . . . 7 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
5756feq3d 6724 . . . . . 6 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆𝐵))
5857anbi1d 631 . . . . 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)))))
59 2ndctop 23471 . . . . . . . . . . . 12 (𝐽 ∈ 2ndω → 𝐽 ∈ Top)
6059adantl 481 . . . . . . . . . . 11 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ Top)
6160ad2antrr 726 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐽 ∈ Top)
62 frn 6744 . . . . . . . . . . . 12 (𝑓:𝑆𝐵 → ran 𝑓𝐵)
6362ad2antrl 728 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐵)
64 bastg 22989 . . . . . . . . . . . . 13 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
6564ad3antrrr 730 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐵 ⊆ (topGen‘𝐵))
66 2ndcctbss.1 . . . . . . . . . . . 12 𝐽 = (topGen‘𝐵)
6765, 66sseqtrrdi 4047 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐵𝐽)
6863, 67sstrd 4006 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐽)
69 simprrl 781 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑜𝐽)
70 simprr 773 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽)) → (topGen‘𝑐) = 𝐽)
7170ad2antlr 727 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → (topGen‘𝑐) = 𝐽)
7269, 71eleqtrrd 2842 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑜 ∈ (topGen‘𝑐))
73 simprrr 782 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑡𝑜)
74 tg2 22988 . . . . . . . . . . . . . 14 ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡𝑜) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
7572, 73, 74syl2anc 584 . . . . . . . . . . . . 13 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
76 bastg 22989 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
7776ad2antrl 728 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
7877ad2antrr 726 . . . . . . . . . . . . . . . . 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))
7966eqeq2i 2748 . . . . . . . . . . . . . . . . . . . . 21 ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
8079biimpi 216 . . . . . . . . . . . . . . . . . . . 20 ((topGen‘𝑐) = 𝐽 → (topGen‘𝑐) = (topGen‘𝐵))
8180adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (topGen‘𝑐) = (topGen‘𝐵))
8281ad2antll 729 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
8382ad2antrr 726 . . . . . . . . . . . . . . . . 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))
8478, 83sseqtrd 4036 . . . . . . . . . . . . . . . 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))
85 simprl 771 . . . . . . . . . . . . . . . 16 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑑𝑐)
8684, 85sseldd 3996 . . . . . . . . . . . . . . 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))
87 simprrl 781 . . . . . . . . . . . . . . 15 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → 𝑡𝑑)
88 tg2 22988 . . . . . . . . . . . . . . 15 ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡𝑑) → ∃𝑚𝐵 (𝑡𝑚𝑚𝑑))
8986, 87, 88syl2anc 584 . . . . . . . . . . . . . 14 (((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) ∧ (𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜))) → ∃𝑚𝐵 (𝑡𝑚𝑚𝑑))
9064ad3antrrr 730 . . . . . . . . . . . . . . . . . . 19 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝐵 ⊆ (topGen‘𝐵))
9190ad2antrr 726 . . . . . . . . . . . . . . . . . 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))
9271ad2antrr 726 . . . . . . . . . . . . . . . . . . 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)
9392, 66eqtr2di 2792 . . . . . . . . . . . . . . . . . 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))
9491, 93sseqtrd 4036 . . . . . . . . . . . . . . . . 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))
95 simprl 771 . . . . . . . . . . . . . . . . 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)
9694, 95sseldd 3996 . . . . . . . . . . . . . . . 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))
97 simprrl 781 . . . . . . . . . . . . . . . 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)
98 tg2 22988 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡𝑚) → ∃𝑛𝑐 (𝑡𝑛𝑛𝑚))
9996, 97, 98syl2anc 584 . . . . . . . . . . . . . . 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))
100 ffn 6737 . . . . . . . . . . . . . . . . . . . 20 (𝑓:𝑆𝐵𝑓 Fn 𝑆)
101100ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → 𝑓 Fn 𝑆)
102101ad2antlr 727 . . . . . . . . . . . . . . . . . 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)
103102ad2antrr 726 . . . . . . . . . . . . . . . . 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)
104 simprl 771 . . . . . . . . . . . . . . . . . 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)
10585ad2antrr 726 . . . . . . . . . . . . . . . . . 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)
106 simplrl 777 . . . . . . . . . . . . . . . . . . 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)
107 simprrr 782 . . . . . . . . . . . . . . . . . . 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)
108 simprr 773 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → 𝑚𝑑)
109108ad2antlr 727 . . . . . . . . . . . . . . . . . . 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)
110 sseq2 4022 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑛𝑤𝑛𝑚))
111 sseq1 4021 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑤𝑑𝑚𝑑))
112110, 111anbi12d 632 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑚 → ((𝑛𝑤𝑤𝑑) ↔ (𝑛𝑚𝑚𝑑)))
113112rspcev 3622 . . . . . . . . . . . . . . . . . . 19 ((𝑚𝐵 ∧ (𝑛𝑚𝑚𝑑)) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
114106, 107, 109, 113syl12anc 837 . . . . . . . . . . . . . . . . . 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))
115 df-br 5149 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
116 vex 3482 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
117 vex 3482 . . . . . . . . . . . . . . . . . . . 20 𝑑 ∈ V
118 simpl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑢 = 𝑛)
119118eleq1d 2824 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑢𝑐𝑛𝑐))
120 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑣 = 𝑑)
121120eleq1d 2824 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑣𝑐𝑑𝑐))
122 sseq1 4021 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑛 → (𝑢𝑤𝑛𝑤))
123 sseq2 4022 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑑 → (𝑤𝑣𝑤𝑑))
124122, 123bi2anan9 638 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑤𝑤𝑣) ↔ (𝑛𝑤𝑤𝑑)))
125124rexbidv 3177 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
126119, 121, 1253anbi123d 1435 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))))
127116, 117, 126, 8braba 5547 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
128115, 127bitr3i 277 . . . . . . . . . . . . . . . . . 18 (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
129104, 105, 114, 128syl3anbrc 1342 . . . . . . . . . . . . . . . . 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)
130 fnfvelrn 7100 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
131103, 129, 130syl2anc 584 . . . . . . . . . . . . . . . 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)
132 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑐)
133 simplll 775 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑑𝑐)
134 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝐵)
135 simprrr 782 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑚)
136108ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝑑)
137134, 135, 136, 113syl12anc 837 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
138132, 133, 137, 128syl3anbrc 1342 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
139 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (1st𝑥) = (1st ‘⟨𝑛, 𝑑⟩))
140 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
141139, 140sseq12d 4029 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((1st𝑥) ⊆ (𝑓𝑥) ↔ (1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
142 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (2nd𝑥) = (2nd ‘⟨𝑛, 𝑑⟩))
143140, 142sseq12d 4029 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓𝑥) ⊆ (2nd𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)))
144141, 143anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑛, 𝑑⟩ → (((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) ↔ ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
145144rspcv 3618 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑛, 𝑑⟩ ∈ 𝑆 → (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
146138, 145syl 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⟩))))
147116, 117op1st 8021 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑛, 𝑑⟩) = 𝑛
148147sseq1i 4024 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
149116, 117op2nd 8022 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑛, 𝑑⟩) = 𝑑
150149sseq2i 4025 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
151148, 150anbi12i 628 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)) ↔ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑))
152 simprl 771 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
153 simprl 771 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → 𝑡𝑛)
154153ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡𝑛)
155152, 154sseldd 3996 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩))
156 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
157 simplrr 778 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑑𝑜)
158157ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑑𝑜)
159156, 158sstrd 4006 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)
160155, 159jca 511 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
161160ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ((𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
162151, 161biimtrid 242 . . . . . . . . . . . . . . . . . . . . 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)))
163146, 162syldc 48 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
164163exp4c 432 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
165164ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
166165adantl 481 . . . . . . . . . . . . . . . . 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)))))
167166imp41 425 . . . . . . . . . . . . . . . 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))
168 eleq2 2828 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡𝑏𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
169 sseq1 4021 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑏𝑜 ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
170168, 169anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → ((𝑡𝑏𝑏𝑜) ↔ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
171170rspcev 3622 . . . . . . . . . . . . . . . 16 (((𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓 ∧ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
172131, 167, 171syl2anc 584 . . . . . . . . . . . . . . 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))
17399, 172rexlimddv 3159 . . . . . . . . . . . . . 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))
17489, 173rexlimddv 3159 . . . . . . . . . . . . 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))
17575, 174rexlimddv 3159 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
176175expr 456 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ((𝑜𝐽𝑡𝑜) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜)))
177176ralrimivv 3198 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ∀𝑜𝐽𝑡𝑜𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
178 basgen2 23012 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ∀𝑜𝐽𝑡𝑜𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜)) → (topGen‘ran 𝑓) = 𝐽)
17961, 68, 177, 178syl3anc 1370 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → (topGen‘ran 𝑓) = 𝐽)
180179, 61eqeltrd 2839 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → (topGen‘ran 𝑓) ∈ Top)
181 tgclb 22993 . . . . . . . 8 (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
182180, 181sylibr 234 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ∈ TopBases)
183 omelon 9684 . . . . . . . . . 10 ω ∈ On
18424adantr 480 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ≼ ω)
185 ondomen 10075 . . . . . . . . . 10 ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
186183, 184, 185sylancr 587 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ∈ dom card)
187100ad2antrl 728 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑓 Fn 𝑆)
188 dffn4 6827 . . . . . . . . . 10 (𝑓 Fn 𝑆𝑓:𝑆onto→ran 𝑓)
189187, 188sylib 218 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑓:𝑆onto→ran 𝑓)
190 fodomnum 10095 . . . . . . . . 9 (𝑆 ∈ dom card → (𝑓:𝑆onto→ran 𝑓 → ran 𝑓𝑆))
191186, 189, 190sylc 65 . . . . . . . 8 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝑆)
192 domtr 9046 . . . . . . . 8 ((ran 𝑓𝑆𝑆 ≼ ω) → ran 𝑓 ≼ ω)
193191, 184, 192syl2anc 584 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ≼ ω)
194179eqcomd 2741 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐽 = (topGen‘ran 𝑓))
195 breq1 5151 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
196 sseq1 4021 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏𝐵 ↔ ran 𝑓𝐵))
197 fveq2 6907 . . . . . . . . . 10 (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
198197eqeq2d 2746 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
199195, 196, 1983anbi123d 1435 . . . . . . . 8 (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))))
200199rspcev 3622 . . . . . . 7 ((ran 𝑓 ∈ TopBases ∧ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
201182, 193, 63, 194, 200syl13anc 1371 . . . . . 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)))
202201ex 412 . . . . 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))))
20358, 202sylbid 240 . . . 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))))
204203exlimdv 1931 . . 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))))
20555, 204mpd 15 . 2 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
2063, 205rexlimddv 3159 1 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  cop 4637   class class class wbr 5148  {copab 5210   I cid 5582   × cxp 5687  dom cdm 5689  ran crn 5690  Rel wrel 5694  Oncon0 6386   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  ωcom 7887  1st c1st 8011  2nd c2nd 8012  cen 8981  cdom 8982  cardccrd 9973  topGenctg 17484  Topctop 22915  TopBasesctb 22968  2ndωc2ndc 23462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977  df-acn 9980  df-topgen 17490  df-top 22916  df-bases 22969  df-2ndc 23464
This theorem is referenced by: (None)
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