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Theorem is2ndc 21970
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 21964 . . 3 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
21eleq2i 2909 . 2 (𝐽 ∈ 2ndω ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)})
3 simpr 485 . . . . 5 ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽)
4 fvex 6680 . . . . 5 (topGen‘𝑥) ∈ V
53, 4syl6eqelr 2927 . . . 4 ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
65rexlimivw 3287 . . 3 (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V)
7 eqeq2 2838 . . . . 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 3302 . . 3 (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)))
106, 9elab3 3678 . 2 (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
112, 10bitri 276 1 (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  {cab 2804  wrex 3144  Vcvv 3500   class class class wbr 5063  cfv 6352  ωcom 7568  cdom 8496  topGenctg 16701  TopBasesctb 21469  2ndωc2ndc 21962
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-sn 4565  df-pr 4567  df-uni 4838  df-iota 6312  df-fv 6360  df-2ndc 21964
This theorem is referenced by:  2ndctop  21971  2ndci  21972  2ndcsb  21973  2ndcredom  21974  2ndc1stc  21975  2ndcrest  21978  2ndcctbss  21979  2ndcdisj  21980  2ndcomap  21982  2ndcsep  21983  dis2ndc  21984  tx2ndc  22175
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