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Theorem is1stc 22049
 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 4814 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 is1stc.1 . . . 4 𝑋 = 𝐽
31, 2eqtr4di 2854 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 pweq 4516 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
5 raleq 3361 . . . . 5 (𝑗 = 𝐽 → (∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))))
65anbi2d 631 . . . 4 (𝑗 = 𝐽 → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))
74, 6rexeqbidv 3358 . . 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 3357 . 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 22047 . 2 1stω = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))}
108, 9elrab2 3634 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110   ∩ cin 3883  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:  is1stc2  22050  1stctop  22051
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