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Theorem is1stc 21768
 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 4716 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 is1stc.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2825 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 pweq 4419 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
5 raleq 3338 . . . . 5 (𝑗 = 𝐽 → (∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))))
65anbi2d 620 . . . 4 (𝑗 = 𝐽 → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
74, 6rexeqbidv 3335 . . 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 3334 . 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 21766 . 2 1st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))}
108, 9elrab2 3592 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 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3081  ∃wrex 3082   ∩ cin 3821  𝒫 cpw 4416  ∪ cuni 4708   class class class wbr 4925  ωcom 7394   ≼ cdom 8302  Topctop 21220  1st𝜔c1stc 21764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-in 3829  df-ss 3836  df-pw 4418  df-uni 4709  df-1stc 21766 This theorem is referenced by:  is1stc2  21769  1stctop  21770
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