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Theorem is2ndc 21230
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 21224 . . 3 2nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
21eleq2i 2691 . 2 (𝐽 ∈ 2nd𝜔 ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)})
3 simpr 477 . . . . 5 ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽)
4 fvex 6188 . . . . 5 (topGen‘𝑥) ∈ V
53, 4syl6eqelr 2708 . . . 4 ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
65rexlimivw 3025 . . 3 (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7 eqeq2 2631 . . . . 5 (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽))
87anbi2d 739 . . . 4 (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
98rexbidv 3048 . . 3 (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
106, 9elab3 3352 . 2 (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
112, 10bitri 264 1 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wrex 2910  Vcvv 3195   class class class wbr 4644  cfv 5876  ωcom 7050  cdom 7938  topGenctg 16079  TopBasesctb 20730  2nd𝜔c2ndc 21222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839  df-fv 5884  df-2ndc 21224
This theorem is referenced by:  2ndctop  21231  2ndci  21232  2ndcsb  21233  2ndcredom  21234  2ndc1stc  21235  2ndcrest  21238  2ndcctbss  21239  2ndcdisj  21240  2ndcomap  21242  2ndcsep  21243  dis2ndc  21244  tx2ndc  21435
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