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Theorem is1stc 21444
Description: The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
is1stc.1 𝑋 = 𝐽
Assertion
Ref Expression
is1stc (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem is1stc
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4594 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 is1stc.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2810 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 pweq 4303 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
5 raleq 3275 . . . . 5 (𝑗 = 𝐽 → (∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))))
65anbi2d 742 . . . 4 (𝑗 = 𝐽 → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
74, 6rexeqbidv 3290 . . 3 (𝑗 = 𝐽 → (∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
83, 7raleqbidv 3289 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
9 df-1stc 21442 . 2 1st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))}
108, 9elrab2 3505 1 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wral 3048  wrex 3049  cin 3712  𝒫 cpw 4300   cuni 4586   class class class wbr 4802  ωcom 7228  cdom 8117  Topctop 20898  1st𝜔c1stc 21440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587  df-1stc 21442
This theorem is referenced by:  is1stc2  21445  1stctop  21446
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