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Theorem is2ndc 22797
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 22791 . . 3 2ndΟ‰ = {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)}
21eleq2i 2829 . 2 (𝐽 ∈ 2ndΟ‰ ↔ 𝐽 ∈ {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)})
3 simpr 485 . . . . 5 ((π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽) β†’ (topGenβ€˜π‘₯) = 𝐽)
4 fvex 6855 . . . . 5 (topGenβ€˜π‘₯) ∈ V
53, 4eqeltrrdi 2846 . . . 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 2748 . . . . 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 3638 . 2 (𝐽 ∈ {𝑗 ∣ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝑗)} ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽))
112, 10bitri 274 1 (𝐽 ∈ 2ndΟ‰ ↔ βˆƒπ‘₯ ∈ TopBases (π‘₯ β‰Ό Ο‰ ∧ (topGenβ€˜π‘₯) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  βˆƒwrex 3073  Vcvv 3445   class class class wbr 5105  β€˜cfv 6496  Ο‰com 7802   β‰Ό cdom 8881  topGenctg 17319  TopBasesctb 22295  2ndΟ‰c2ndc 22789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rex 3074  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-sn 4587  df-pr 4589  df-uni 4866  df-iota 6448  df-fv 6504  df-2ndc 22791
This theorem is referenced by:  2ndctop  22798  2ndci  22799  2ndcsb  22800  2ndcredom  22801  2ndc1stc  22802  2ndcrest  22805  2ndcctbss  22806  2ndcdisj  22807  2ndcomap  22809  2ndcsep  22810  dis2ndc  22811  tx2ndc  23002
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