Theorem is2ndc 23301

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 23295	. . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
2	1	eleq2i 2819	. 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 6897	. . . . 5 ⊢ (topGen‘𝑥) ∈ V
5	3, 4	eqeltrrdi 2836	. . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
6	5	rexlimivw 3145	. . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7		eqeq2 2738	. . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽))
8	7	anbi2d 628	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
9	8	rexbidv 3172	. . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
10	6, 9	elab3 3671	. 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 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  ∃wrex 3064  Vcvv 3468   class class class wbr 5141  ‘cfv 6536  ωcom 7851   ≼ cdom 8936  topGenctg 17390  TopBasesctb 22799  2^ndωc2ndc 23293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rex 3065  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6488  df-fv 6544  df-2ndc 23295

This theorem is referenced by:  2ndctop  23302  2ndci  23303  2ndcsb  23304  2ndcredom  23305  2ndc1stc  23306  2ndcrest  23309  2ndcctbss  23310  2ndcdisj  23311  2ndcomap  23313  2ndcsep  23314  dis2ndc  23315  tx2ndc  23506