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Theorem is1stc2 22050
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 22049 . 2 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
3 elin 3900 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤 ∈ 𝒫 𝑧))
4 velpw 4505 . . . . . . . . . . . . . 14 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
54anbi2i 625 . . . . . . . . . . . . 13 ((𝑤𝑦𝑤 ∈ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
63, 5bitri 278 . . . . . . . . . . . 12 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
76anbi2i 625 . . . . . . . . . . 11 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)))
8 an12 644 . . . . . . . . . . 11 ((𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
97, 8bitri 278 . . . . . . . . . 10 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
109exbii 1849 . . . . . . . . 9 (∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
11 eluni 4806 . . . . . . . . 9 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12 df-rex 3115 . . . . . . . . 9 (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
1310, 11, 123bitr4i 306 . . . . . . . 8 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))
1413imbi2i 339 . . . . . . 7 ((𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1514ralbii 3136 . . . . . 6 (∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1615anbi2i 625 . . . . 5 ((𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1716rexbii 3213 . . . 4 (∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1817ralbii 3136 . . 3 (∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1918anbi2i 625 . 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 278 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 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  cin 3883  wss 3884  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  ωcom 7564  cdom 8494  Topctop 21501  1stωc1stc 22045
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804  df-1stc 22047
This theorem is referenced by:  1stcclb  22052  2ndc1stc  22059  1stcrest  22061  lly1stc  22104  tx1stc  22258  met1stc  23131
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