Theorem 2ndci 22048

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 485	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ∈ TopBases)
2		simpr 487	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
3		eqidd 2820	. . 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)))
6	4, 5	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)) ↔ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))))
7	6	rspcev 3621	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ (𝐵 ≼ ω ∧ (topGen‘𝐵) = (topGen‘𝐵))) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
8	1, 2, 3, 7	syl12anc 834	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
9		is2ndc 22046	. 2 ⊢ ((topGen‘𝐵) ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = (topGen‘𝐵)))
10	8, 9	sylibr 236	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∃wrex 3137   class class class wbr 5057  ‘cfv 6348  ωcom 7572   ≼ cdom 8499  topGenctg 16703  TopBasesctb 21545  2^ndωc2ndc 22038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-2ndc 22040

This theorem is referenced by:  2ndcrest  22054  2ndcomap  22058  dis2ndc  22060  dis1stc  22099  tx2ndc  22251  met2ndci  23124  re2ndc  23401