Theorem 2ndci 23642

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 2767	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) = (topGen‘𝐵))
4		breq1 5117	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω))
5		fveqeq2 6897	. . . . 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 3584	. . 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 23640	. 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 2146  ∃wrex 3092   class class class wbr 5114  ‘cfv 6543  ωcom 7871   ≼ cdom 8950  topGenctg 17515  TopBasesctb 23139  2^ndωc2ndc 23632

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-2ndc 23634

This theorem is used by:  2ndcrest  23648  2ndcomap  23652  dis2ndc  23654  dis1stc  23693  tx2ndc  23845  met2ndci  24716  re2ndc  24995