Theorem is2ndc 23455

Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
is2ndc	⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))

Distinct variable group:   𝑥,𝐽

Proof of Theorem is2ndc

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-2ndc 23449	. . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
2	1	eleq2i 2832	. 2 ⊢ (𝐽 ∈ 2ndω ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)})
3		simpr 484	. . . . 5 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽)
4		fvex 6918	. . . . 5 ⊢ (topGen‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2849	. . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
6	5	rexlimivw 3150	. . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7		eqeq2 2748	. . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽))
8	7	anbi2d 630	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
9	8	rexbidv 3178	. . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
10	6, 9	elab3 3685	. 2 ⊢ (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
11	2, 10	bitri 275	1 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  ∃wrex 3069  Vcvv 3479   class class class wbr 5142  ‘cfv 6560  ωcom 7888   ≼ cdom 8984  topGenctg 17483  TopBasesctb 22953  2^ndωc2ndc 23447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513  df-fv 6568  df-2ndc 23449

This theorem is referenced by:  2ndctop  23456  2ndci  23457  2ndcsb  23458  2ndcredom  23459  2ndc1stc  23460  2ndcrest  23463  2ndcctbss  23464  2ndcdisj  23465  2ndcomap  23467  2ndcsep  23468  dis2ndc  23469  tx2ndc  23660