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Theorem is1stc2 23297
Description: An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
is1stc.1 𝑋 = 𝐽
Assertion
Ref Expression
is1stc2 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋
Allowed substitution hints:   𝐽(𝑤)   𝑋(𝑦,𝑧,𝑤)

Proof of Theorem is1stc2
StepHypRef Expression
1 is1stc.1 . . 3 𝑋 = 𝐽
21is1stc 23296 . 2 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
3 elin 3959 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤 ∈ 𝒫 𝑧))
4 velpw 4602 . . . . . . . . . . . . . 14 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
54anbi2i 622 . . . . . . . . . . . . 13 ((𝑤𝑦𝑤 ∈ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
63, 5bitri 275 . . . . . . . . . . . 12 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
76anbi2i 622 . . . . . . . . . . 11 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)))
8 an12 642 . . . . . . . . . . 11 ((𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
97, 8bitri 275 . . . . . . . . . 10 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
109exbii 1842 . . . . . . . . 9 (∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
11 eluni 4905 . . . . . . . . 9 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12 df-rex 3065 . . . . . . . . 9 (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
1310, 11, 123bitr4i 303 . . . . . . . 8 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))
1413imbi2i 336 . . . . . . 7 ((𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1514ralbii 3087 . . . . . 6 (∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1615anbi2i 622 . . . . 5 ((𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1716rexbii 3088 . . . 4 (∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1817ralbii 3087 . . 3 (∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1918anbi2i 622 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))) ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
202, 19bitri 275 1 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wral 3055  wrex 3064  cin 3942  wss 3943  𝒫 cpw 4597   cuni 4902   class class class wbr 5141  ωcom 7851  cdom 8936  Topctop 22746  1stωc1stc 23292
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903  df-1stc 23294
This theorem is referenced by:  1stcclb  23299  2ndc1stc  23306  1stcrest  23308  lly1stc  23351  tx1stc  23505  met1stc  24381
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