Theorem 2ndci 23821

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.

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ∈ TopBases)
2		simpr 490	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
3		eqidd 2764	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) = (topGen‘𝐵))
4		breq1 5111	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω))
5		fveqeq2 6925	. . . . 5 ⊢ (𝑥 = 𝐵 → ((topGen‘𝑥) = (topGen‘𝐵) ↔ (topGen‘𝐵) = (topGen‘𝐵)))
6	4, 5	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)) ↔ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))))
7	6	rspcev 3580	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
8	1, 2, 3, 7	syl12anc 850	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
9		is2ndc 23819	. 2 ⊢ ((topGen‘𝐵) ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
10	8, 9	sylibr 237	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5108  ‘cfv 6566  ωcom 7911   ≼ cdom 9009  topGenctg 17658  TopBasesctb 23318  2^ndωc2ndc 23811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-2ndc 23813

This theorem is used by:  2ndcrest  23827  2ndcomap  23832  dis2ndc  23834  dis1stc  23873  tx2ndc  24025  met2ndci  24896  re2ndc  25175