Theorem is2ndc 23340

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 23334	. . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
2	1	eleq2i 2821	. 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 6874	. . . . 5 ⊢ (topGen‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2838	. . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
6	5	rexlimivw 3131	. . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7		eqeq2 2742	. . . . 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 3158	. . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
10	6, 9	elab3 3656	. 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 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450   class class class wbr 5110  ‘cfv 6514  ωcom 7845   ≼ cdom 8919  topGenctg 17407  TopBasesctb 22839  2^ndωc2ndc 23332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rex 3055  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-uni 4875  df-iota 6467  df-fv 6522  df-2ndc 23334

This theorem is referenced by:  2ndctop  23341  2ndci  23342  2ndcsb  23343  2ndcredom  23344  2ndc1stc  23345  2ndcrest  23348  2ndcctbss  23349  2ndcdisj  23350  2ndcomap  23352  2ndcsep  23353  dis2ndc  23354  tx2ndc  23545