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Theorem 2ndcctbss 21991
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 485 . . 3 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ 2ndω)
2 is2ndc 21982 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
31, 2sylib 219 . 2 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑐 ∈ TopBases (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))
4 vex 3495 . . . . . . 7 𝑐 ∈ V
54, 4xpex 7465 . . . . . 6 (𝑐 × 𝑐) ∈ V
6 3simpa 1140 . . . . . . . 8 ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) → (𝑢𝑐𝑣𝑐))
76ssopab2i 5428 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
8 2ndcctbss.2 . . . . . . 7 𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}
9 df-xp 5554 . . . . . . 7 (𝑐 × 𝑐) = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐)}
107, 8, 93sstr4i 4007 . . . . . 6 𝑆 ⊆ (𝑐 × 𝑐)
11 ssdomg 8543 . . . . . 6 ((𝑐 × 𝑐) ∈ V → (𝑆 ⊆ (𝑐 × 𝑐) → 𝑆 ≼ (𝑐 × 𝑐)))
125, 10, 11mp2 9 . . . . 5 𝑆 ≼ (𝑐 × 𝑐)
134xpdom1 8604 . . . . . . . . 9 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ (ω × 𝑐))
14 omex 9094 . . . . . . . . . 10 ω ∈ V
1514xpdom2 8600 . . . . . . . . 9 (𝑐 ≼ ω → (ω × 𝑐) ≼ (ω × ω))
16 domtr 8550 . . . . . . . . 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 9429 . . . . . . . 8 (ω × ω) ≈ ω
19 domentr 8556 . . . . . . . 8 (((𝑐 × 𝑐) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑐 × 𝑐) ≼ ω)
2017, 18, 19sylancl 586 . . . . . . 7 (𝑐 ≼ ω → (𝑐 × 𝑐) ≼ ω)
2120adantr 481 . . . . . 6 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (𝑐 × 𝑐) ≼ ω)
2221ad2antll 725 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑐 × 𝑐) ≼ ω)
23 domtr 8550 . . . . 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)
258relopabi 5687 . . . . . . . . 9 Rel 𝑆
26 simpr 485 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → 𝑥𝑆)
27 1st2nd 7727 . . . . . . . . 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 2911 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
30 df-br 5058 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆)
31 fvex 6676 . . . . . . . . . 10 (1st𝑥) ∈ V
32 fvex 6676 . . . . . . . . . 10 (2nd𝑥) ∈ V
33 simpl 483 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑢 = (1st𝑥))
3433eleq1d 2894 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑢𝑐 ↔ (1st𝑥) ∈ 𝑐))
35 simpr 485 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → 𝑣 = (2nd𝑥))
3635eleq1d 2894 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (𝑣𝑐 ↔ (2nd𝑥) ∈ 𝑐))
37 sseq1 3989 . . . . . . . . . . . . 13 (𝑢 = (1st𝑥) → (𝑢𝑤 ↔ (1st𝑥) ⊆ 𝑤))
38 sseq2 3990 . . . . . . . . . . . . 13 (𝑣 = (2nd𝑥) → (𝑤𝑣𝑤 ⊆ (2nd𝑥)))
3937, 38bi2anan9 635 . . . . . . . . . . . 12 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑤𝑤𝑣) ↔ ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4039rexbidv 3294 . . . . . . . . . . 11 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4134, 36, 403anbi123d 1427 . . . . . . . . . 10 ((𝑢 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))))
4231, 32, 41, 8braba 5415 . . . . . . . . 9 ((1st𝑥)𝑆(2nd𝑥) ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4330, 42bitr3i 278 . . . . . . . 8 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑆 ↔ ((1st𝑥) ∈ 𝑐 ∧ (2nd𝑥) ∈ 𝑐 ∧ ∃𝑤𝐵 ((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))))
4443simp3bi 1139 . . . . . . 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 6733 . . . . . . . 8 (𝐵 ∈ TopBases → ( I ‘𝐵) = 𝐵)
4746ad3antrrr 726 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ( I ‘𝐵) = 𝐵)
4847rexeqdv 3414 . . . . . 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 258 . . . . 5 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ 𝑥𝑆) → ∃𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
5049ralrimiva 3179 . . . 4 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)))
51 fvex 6676 . . . . 5 ( I ‘𝐵) ∈ V
52 sseq2 3990 . . . . . 6 (𝑤 = (𝑓𝑥) → ((1st𝑥) ⊆ 𝑤 ↔ (1st𝑥) ⊆ (𝑓𝑥)))
53 sseq1 3989 . . . . . 6 (𝑤 = (𝑓𝑥) → (𝑤 ⊆ (2nd𝑥) ↔ (𝑓𝑥) ⊆ (2nd𝑥)))
5452, 53anbi12d 630 . . . . 5 (𝑤 = (𝑓𝑥) → (((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥)) ↔ ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5551, 54axcc4dom 9851 . . . 4 ((𝑆 ≼ ω ∧ ∀𝑥𝑆𝑤 ∈ ( I ‘𝐵)((1st𝑥) ⊆ 𝑤𝑤 ⊆ (2nd𝑥))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5624, 50, 55syl2anc 584 . . 3 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ∃𝑓(𝑓:𝑆⟶( I ‘𝐵) ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))))
5746ad2antrr 722 . . . . . . 7 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → ( I ‘𝐵) = 𝐵)
5857feq3d 6494 . . . . . 6 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (𝑓:𝑆⟶( I ‘𝐵) ↔ 𝑓:𝑆𝐵))
5958anbi1d 629 . . . . 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 21983 . . . . . . . . . . . 12 (𝐽 ∈ 2ndω → 𝐽 ∈ Top)
6160adantl 482 . . . . . . . . . . 11 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → 𝐽 ∈ Top)
6261ad2antrr 722 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐽 ∈ Top)
63 frn 6513 . . . . . . . . . . . 12 (𝑓:𝑆𝐵 → ran 𝑓𝐵)
6463ad2antrl 724 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐵)
65 bastg 21502 . . . . . . . . . . . . 13 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
6665ad3antrrr 726 . . . . . . . . . . . 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, 67sseqtrrdi 4015 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝐵𝐽)
6964, 68sstrd 3974 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓𝐽)
70 simprrl 777 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑜𝐽)
71 simprr 769 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽)) → (topGen‘𝑐) = 𝐽)
7271ad2antlr 723 . . . . . . . . . . . . . . 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 2913 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝑡𝑜)
75 tg2 21501 . . . . . . . . . . . . . 14 ((𝑜 ∈ (topGen‘𝑐) ∧ 𝑡𝑜) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
7673, 74, 75syl2anc 584 . . . . . . . . . . . . 13 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → ∃𝑑𝑐 (𝑡𝑑𝑑𝑜))
77 bastg 21502 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ TopBases → 𝑐 ⊆ (topGen‘𝑐))
7877ad2antrl 724 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → 𝑐 ⊆ (topGen‘𝑐))
7978ad2antrr 722 . . . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . . . . . . . 21 ((topGen‘𝑐) = 𝐽 ↔ (topGen‘𝑐) = (topGen‘𝐵))
8180biimpi 217 . . . . . . . . . . . . . . . . . . . 20 ((topGen‘𝑐) = 𝐽 → (topGen‘𝑐) = (topGen‘𝐵))
8281adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽) → (topGen‘𝑐) = (topGen‘𝐵))
8382ad2antll 725 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) → (topGen‘𝑐) = (topGen‘𝐵))
8483ad2antrr 722 . . . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 3965 . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 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 21501 . . . . . . . . . . . . . . 15 ((𝑑 ∈ (topGen‘𝐵) ∧ 𝑡𝑑) → ∃𝑚𝐵 (𝑡𝑚𝑚𝑑))
9087, 88, 89syl2anc 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))
9165ad3antrrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ ((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜))) → 𝐵 ⊆ (topGen‘𝐵))
9291ad2antrr 722 . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . 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 2870 . . . . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 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 3965 . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . 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 21501 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (topGen‘𝑐) ∧ 𝑡𝑚) → ∃𝑛𝑐 (𝑡𝑛𝑛𝑚))
10097, 98, 99syl2anc 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))
101 ffn 6507 . . . . . . . . . . . . . . . . . . . 20 (𝑓:𝑆𝐵𝑓 Fn 𝑆)
102101ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → 𝑓 Fn 𝑆)
103102ad2antlr 723 . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → 𝑚𝑑)
110109ad2antlr 723 . . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑛𝑤𝑛𝑚))
112 sseq1 3989 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑚 → (𝑤𝑑𝑚𝑑))
113111, 112anbi12d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑚 → ((𝑛𝑤𝑤𝑑) ↔ (𝑛𝑚𝑚𝑑)))
114113rspcev 3620 . . . . . . . . . . . . . . . . . . 19 ((𝑚𝐵 ∧ (𝑛𝑚𝑚𝑑)) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
115107, 108, 110, 114syl12anc 832 . . . . . . . . . . . . . . . . . 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 5058 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ ⟨𝑛, 𝑑⟩ ∈ 𝑆)
117 vex 3495 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
118 vex 3495 . . . . . . . . . . . . . . . . . . . 20 𝑑 ∈ V
119 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑢 = 𝑛)
120119eleq1d 2894 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑢𝑐𝑛𝑐))
121 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → 𝑣 = 𝑑)
122121eleq1d 2894 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (𝑣𝑐𝑑𝑐))
123 sseq1 3989 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑛 → (𝑢𝑤𝑛𝑤))
124 sseq2 3990 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑑 → (𝑤𝑣𝑤𝑑))
125123, 124bi2anan9 635 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑤𝑤𝑣) ↔ (𝑛𝑤𝑤𝑑)))
126125rexbidv 3294 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 = 𝑛𝑣 = 𝑑) → (∃𝑤𝐵 (𝑢𝑤𝑤𝑣) ↔ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
127120, 122, 1263anbi123d 1427 . . . . . . . . . . . . . . . . . . . 20 ((𝑢 = 𝑛𝑣 = 𝑑) → ((𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣)) ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))))
128117, 118, 127, 8braba 5415 . . . . . . . . . . . . . . . . . . 19 (𝑛𝑆𝑑 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
129116, 128bitr3i 278 . . . . . . . . . . . . . . . . . 18 (⟨𝑛, 𝑑⟩ ∈ 𝑆 ↔ (𝑛𝑐𝑑𝑐 ∧ ∃𝑤𝐵 (𝑛𝑤𝑤𝑑)))
130105, 106, 115, 129syl3anbrc 1335 . . . . . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑆 ∧ ⟨𝑛, 𝑑⟩ ∈ 𝑆) → (𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓)
132104, 130, 131syl2anc 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)
133 simprl 767 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑐)
134 simplll 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑑𝑐)
135 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝐵)
136 simprrr 778 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑛𝑚)
137109ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → 𝑚𝑑)
138135, 136, 137, 114syl12anc 832 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ∃𝑤𝐵 (𝑛𝑤𝑤𝑑))
139133, 134, 138, 129syl3anbrc 1335 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) → ⟨𝑛, 𝑑⟩ ∈ 𝑆)
140 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (1st𝑥) = (1st ‘⟨𝑛, 𝑑⟩))
141 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (𝑓𝑥) = (𝑓‘⟨𝑛, 𝑑⟩))
142140, 141sseq12d 3997 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((1st𝑥) ⊆ (𝑓𝑥) ↔ (1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩)))
143 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑛, 𝑑⟩ → (2nd𝑥) = (2nd ‘⟨𝑛, 𝑑⟩))
144141, 143sseq12d 3997 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑛, 𝑑⟩ → ((𝑓𝑥) ⊆ (2nd𝑥) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩)))
145142, 144anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑛, 𝑑⟩ → (((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) ↔ ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩))))
146145rspcv 3615 . . . . . . . . . . . . . . . . . . . . . 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 7686 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑛, 𝑑⟩) = 𝑛
149148sseq1i 3992 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st ‘⟨𝑛, 𝑑⟩) ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ↔ 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
150117, 118op2nd 7687 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑛, 𝑑⟩) = 𝑑
151150sseq2i 3993 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓‘⟨𝑛, 𝑑⟩) ⊆ (2nd ‘⟨𝑛, 𝑑⟩) ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
152149, 151anbi12i 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 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩))
154 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → 𝑡𝑛)
155154ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡𝑛)
156153, 155sseldd 3965 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩))
157 simprr 769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)
158 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) → 𝑑𝑜)
159158ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → 𝑑𝑜)
160157, 159sstrd 3974 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)
161156, 160jca 512 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) ∧ (𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑))) ∧ (𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚))) ∧ (𝑛 ⊆ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑑)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
162161ex 413 . . . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . . . 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 433 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
166165ad2antlr 723 . . . . . . . . . . . . . . . . . 18 (((𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥))) ∧ (𝑜𝐽𝑡𝑜)) → ((𝑑𝑐 ∧ (𝑡𝑑𝑑𝑜)) → ((𝑚𝐵 ∧ (𝑡𝑚𝑚𝑑)) → ((𝑛𝑐 ∧ (𝑡𝑛𝑛𝑚)) → (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))))
167166adantl 482 . . . . . . . . . . . . . . . . 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 426 . . . . . . . . . . . . . . . 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 2898 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑡𝑏𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩)))
170 sseq1 3989 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → (𝑏𝑜 ↔ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜))
171169, 170anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑓‘⟨𝑛, 𝑑⟩) → ((𝑡𝑏𝑏𝑜) ↔ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)))
172171rspcev 3620 . . . . . . . . . . . . . . . 16 (((𝑓‘⟨𝑛, 𝑑⟩) ∈ ran 𝑓 ∧ (𝑡 ∈ (𝑓‘⟨𝑛, 𝑑⟩) ∧ (𝑓‘⟨𝑛, 𝑑⟩) ⊆ 𝑜)) → ∃𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜))
173132, 168, 172syl2anc 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))
174100, 173rexlimddv 3288 . . . . . . . . . . . . . 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 3288 . . . . . . . . . . . . 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 3288 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 3187 . . . . . . . . . 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 21525 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ∀𝑜𝐽𝑡𝑜𝑏 ∈ ran 𝑓(𝑡𝑏𝑏𝑜)) → (topGen‘ran 𝑓) = 𝐽)
18062, 69, 178, 179syl3anc 1363 . . . . . . . . 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 2910 . . . . . . . 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 21506 . . . . . . . 8 (ran 𝑓 ∈ TopBases ↔ (topGen‘ran 𝑓) ∈ Top)
183181, 182sylibr 235 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ∈ TopBases)
184 omelon 9097 . . . . . . . . . 10 ω ∈ On
18524adantr 481 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ≼ ω)
186 ondomen 9451 . . . . . . . . . 10 ((ω ∈ On ∧ 𝑆 ≼ ω) → 𝑆 ∈ dom card)
187184, 185, 186sylancr 587 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑆 ∈ dom card)
188101ad2antrl 724 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → 𝑓 Fn 𝑆)
189 dffn4 6589 . . . . . . . . . 10 (𝑓 Fn 𝑆𝑓:𝑆onto→ran 𝑓)
190188, 189sylib 219 . . . . . . . . 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 9471 . . . . . . . . 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 8550 . . . . . . . 8 ((ran 𝑓𝑆𝑆 ≼ ω) → ran 𝑓 ≼ ω)
194192, 185, 193syl2anc 584 . . . . . . 7 ((((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) ∧ (𝑐 ∈ TopBases ∧ (𝑐 ≼ ω ∧ (topGen‘𝑐) = 𝐽))) ∧ (𝑓:𝑆𝐵 ∧ ∀𝑥𝑆 ((1st𝑥) ⊆ (𝑓𝑥) ∧ (𝑓𝑥) ⊆ (2nd𝑥)))) → ran 𝑓 ≼ ω)
195180eqcomd 2824 . . . . . . 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 5060 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏 ≼ ω ↔ ran 𝑓 ≼ ω))
197 sseq1 3989 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝑏𝐵 ↔ ran 𝑓𝐵))
198 fveq2 6663 . . . . . . . . . 10 (𝑏 = ran 𝑓 → (topGen‘𝑏) = (topGen‘ran 𝑓))
199198eqeq2d 2829 . . . . . . . . 9 (𝑏 = ran 𝑓 → (𝐽 = (topGen‘𝑏) ↔ 𝐽 = (topGen‘ran 𝑓)))
200196, 197, 1993anbi123d 1427 . . . . . . . 8 (𝑏 = ran 𝑓 → ((𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)) ↔ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))))
201200rspcev 3620 . . . . . . 7 ((ran 𝑓 ∈ TopBases ∧ (ran 𝑓 ≼ ω ∧ ran 𝑓𝐵𝐽 = (topGen‘ran 𝑓))) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
202183, 194, 64, 195, 201syl13anc 1364 . . . . . 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 413 . . . . 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 241 . . . 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 1925 . . 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 3288 1 ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2ndω) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933  cop 4563   class class class wbr 5057  {copab 5119   I cid 5452   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553  Oncon0 6184   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  ωcom 7569  1st c1st 7676  2nd c2nd 7677  cen 8494  cdom 8495  cardccrd 9352  topGenctg 16699  Topctop 21429  TopBasesctb 21481  2ndωc2ndc 21974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356  df-acn 9359  df-topgen 16705  df-top 21430  df-bases 21482  df-2ndc 21976
This theorem is referenced by: (None)
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