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Theorem is1stc2 21155
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 21154 . 2 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
3 elin 3774 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤 ∈ 𝒫 𝑧))
4 selpw 4137 . . . . . . . . . . . . . 14 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
54anbi2i 729 . . . . . . . . . . . . 13 ((𝑤𝑦𝑤 ∈ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
63, 5bitri 264 . . . . . . . . . . . 12 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
76anbi2i 729 . . . . . . . . . . 11 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)))
8 an12 837 . . . . . . . . . . 11 ((𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
97, 8bitri 264 . . . . . . . . . 10 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
109exbii 1771 . . . . . . . . 9 (∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
11 eluni 4405 . . . . . . . . 9 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12 df-rex 2913 . . . . . . . . 9 (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
1310, 11, 123bitr4i 292 . . . . . . . 8 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))
1413imbi2i 326 . . . . . . 7 ((𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1514ralbii 2974 . . . . . 6 (∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1615anbi2i 729 . . . . 5 ((𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1716rexbii 3034 . . . 4 (∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1817ralbii 2974 . . 3 (∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1918anbi2i 729 . 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 264 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 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  ωcom 7012  cdom 7897  Topctop 20617  1st𝜔c1stc 21150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-1stc 21152
This theorem is referenced by:  1stcclb  21157  2ndc1stc  21164  1stcrest  21166  lly1stc  21209  tx1stc  21363  met1stc  22236
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