Theorem 2ndci 21251

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 473	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ∈ TopBases)
2		simpr 477	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
3		eqidd 2623	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) = (topGen‘𝐵))
4		breq1 4656	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω))
5		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝐵 → (topGen‘𝑥) = (topGen‘𝐵))
6	5	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = 𝐵 → ((topGen‘𝑥) = (topGen‘𝐵) ↔ (topGen‘𝐵) = (topGen‘𝐵)))
7	4, 6	anbi12d 747	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)) ↔ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))))
8	7	rspcev 3309	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
9	1, 2, 3, 8	syl12anc 1324	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
10		is2ndc 21249	. 2 ⊢ ((topGen‘𝐵) ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
11	9, 10	sylibr 224	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2nd𝜔)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  ωcom 7065   ≼ cdom 7953  topGenctg 16098  TopBasesctb 20749  2^nd𝜔c2ndc 21241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-2ndc 21243

This theorem is referenced by:  2ndcrest  21257  2ndcomap  21261  dis2ndc  21263  dis1stc  21302  tx2ndc  21454  met2ndci  22327  re2ndc  22604