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Theorem is2ndc 23363
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 ∈ 2ndΟ‰ ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽))
Distinct variable group:   π‘₯,𝐽

Proof of Theorem is2ndc
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-2ndc 23357 . . 3 2ndΟ‰ = {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)}
21eleq2i 2821 . 2 (𝐽 ∈ 2ndΟ‰ ↔ 𝐽 ∈ {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)})
3 simpr 484 . . . . 5 ((π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽) β†’ (topGenβ€˜π‘₯) = 𝐽)
4 fvex 6910 . . . . 5 (topGenβ€˜π‘₯) ∈ V
53, 4eqeltrrdi 2838 . . . 4 ((π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽) β†’ 𝐽 ∈ V)
65rexlimivw 3148 . . 3 (βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽) β†’ 𝐽 ∈ V)
7 eqeq2 2740 . . . . 5 (𝑗 = 𝐽 β†’ ((topGenβ€˜π‘₯) = 𝑗 ↔ (topGenβ€˜π‘₯) = 𝐽))
87anbi2d 629 . . . 4 (𝑗 = 𝐽 β†’ ((π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗) ↔ (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽)))
98rexbidv 3175 . . 3 (𝑗 = 𝐽 β†’ (βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗) ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽)))
106, 9elab3 3675 . 2 (𝐽 ∈ {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)} ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽))
112, 10bitri 275 1 (𝐽 ∈ 2ndΟ‰ ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  βˆƒwrex 3067  Vcvv 3471   class class class wbr 5148  β€˜cfv 6548  Ο‰com 7870   β‰Ό cdom 8962  topGenctg 17419  TopBasesctb 22861  2ndΟ‰c2ndc 23355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rex 3068  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4909  df-iota 6500  df-fv 6556  df-2ndc 23357
This theorem is referenced by:  2ndctop  23364  2ndci  23365  2ndcsb  23366  2ndcredom  23367  2ndc1stc  23368  2ndcrest  23371  2ndcctbss  23372  2ndcdisj  23373  2ndcomap  23375  2ndcsep  23376  dis2ndc  23377  tx2ndc  23568
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