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Theorem is2ndc 23270
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.
StepHypRef Expression
1 df-2ndc 23264 . . 3 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
21eleq2i 2824 . 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 6904 . . . . 5 (topGen‘𝑥) ∈ V
53, 4eqeltrrdi 2841 . . . 4 ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
65rexlimivw 3150 . . 3 (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7 eqeq2 2743 . . . . 5 (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽))
87anbi2d 628 . . . 4 (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
98rexbidv 3177 . . 3 (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
106, 9elab3 3676 . 2 (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
112, 10bitri 275 1 (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wrex 3069  Vcvv 3473   class class class wbr 5148  cfv 6543  ωcom 7859  cdom 8943  topGenctg 17390  TopBasesctb 22768  2ndωc2ndc 23262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rex 3070  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551  df-2ndc 23264
This theorem is referenced by:  2ndctop  23271  2ndci  23272  2ndcsb  23273  2ndcredom  23274  2ndc1stc  23275  2ndcrest  23278  2ndcctbss  23279  2ndcdisj  23280  2ndcomap  23282  2ndcsep  23283  dis2ndc  23284  tx2ndc  23475
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