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Theorem is1stc2 21055
 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 21054 . 2 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
3 elin 3758 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤 ∈ 𝒫 𝑧))
4 selpw 4115 . . . . . . . . . . . . . 14 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
54anbi2i 726 . . . . . . . . . . . . 13 ((𝑤𝑦𝑤 ∈ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
63, 5bitri 263 . . . . . . . . . . . 12 (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤𝑦𝑤𝑧))
76anbi2i 726 . . . . . . . . . . 11 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)))
8 an12 834 . . . . . . . . . . 11 ((𝑥𝑤 ∧ (𝑤𝑦𝑤𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
97, 8bitri 263 . . . . . . . . . 10 ((𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
109exbii 1764 . . . . . . . . 9 (∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
11 eluni 4375 . . . . . . . . 9 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥𝑤𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12 df-rex 2902 . . . . . . . . 9 (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤(𝑤𝑦 ∧ (𝑥𝑤𝑤𝑧)))
1310, 11, 123bitr4i 291 . . . . . . . 8 (𝑥 (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))
1413imbi2i 325 . . . . . . 7 ((𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1514ralbii 2963 . . . . . 6 (∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))
1615anbi2i 726 . . . . 5 ((𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1716rexbii 3023 . . . 4 (∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1817ralbii 2963 . . 3 (∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1918anbi2i 726 . 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 263 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  Topctop 20517  1st𝜔c1stc 21050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-1stc 21052 This theorem is referenced by:  1stcclb  21057  2ndc1stc  21064  1stcrest  21066  lly1stc  21109  tx1stc  21263  met1stc  22136
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