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Theorem 2ndci 21471
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 468 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ∈ TopBases)
2 simpr 471 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
3 eqidd 2771 . . 3 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) = (topGen‘𝐵))
4 breq1 4787 . . . . 5 (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω))
5 fveq2 6332 . . . . . 6 (𝑥 = 𝐵 → (topGen‘𝑥) = (topGen‘𝐵))
65eqeq1d 2772 . . . . 5 (𝑥 = 𝐵 → ((topGen‘𝑥) = (topGen‘𝐵) ↔ (topGen‘𝐵) = (topGen‘𝐵)))
74, 6anbi12d 608 . . . 4 (𝑥 = 𝐵 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)) ↔ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))))
87rspcev 3458 . . 3 ((𝐵 ∈ TopBases ∧ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
91, 2, 3, 8syl12anc 1473 . 2 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
10 is2ndc 21469 . 2 ((topGen‘𝐵) ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
119, 10sylibr 224 1 ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wrex 3061   class class class wbr 4784  cfv 6031  ωcom 7211  cdom 8106  topGenctg 16305  TopBasesctb 20969  2nd𝜔c2ndc 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-2ndc 21463
This theorem is referenced by:  2ndcrest  21477  2ndcomap  21481  dis2ndc  21483  dis1stc  21522  tx2ndc  21674  met2ndci  22546  re2ndc  22823
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