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Theorem 2ndci 21984
Description: A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndci ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)

Proof of Theorem 2ndci
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ∈ TopBases)
2 simpr 485 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
3 eqidd 2819 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) = (topGen‘𝐵))
4 breq1 5060 . . . . 5 (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω))
5 fveqeq2 6672 . . . . 5 (𝑥 = 𝐵 → ((topGen‘𝑥) = (topGen‘𝐵) ↔ (topGen‘𝐵) = (topGen‘𝐵)))
64, 5anbi12d 630 . . . 4 (𝑥 = 𝐵 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)) ↔ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))))
76rspcev 3620 . . 3 ((𝐵 ∈ TopBases ∧ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
81, 2, 3, 7syl12anc 832 . 2 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
9 is2ndc 21982 . 2 ((topGen‘𝐵) ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
108, 9sylibr 235 1 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cfv 6348  ωcom 7569  cdom 8495  topGenctg 16699  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-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-2ndc 21976
This theorem is referenced by:  2ndcrest  21990  2ndcomap  21994  dis2ndc  21996  dis1stc  22035  tx2ndc  22187  met2ndci  23059  re2ndc  23336
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