Theorem is1stc2 23429

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 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))

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

Step	Hyp	Ref	Expression
1		is1stc.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	is1stc 23428	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3		elin 3962	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝒫 𝑧))
4		velpw 4611	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
5	4	anbi2i 621	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧))
6	3, 5	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧))
7	6	anbi2i 621	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥 ∈ 𝑤 ∧ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧)))
8		an12 643	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑤 ∧ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
9	7, 8	bitri 274	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
10	9	exbii 1842	. . . . . . . . 9 ⊢ (∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
11		eluni 4915	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12		df-rex 3060	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
13	10, 11, 12	3bitr4i 302	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
14	13	imbi2i 335	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
15	14	ralbii 3082	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
16	15	anbi2i 621	. . . . 5 ⊢ ((𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))) ↔ (𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
17	16	rexbii 3083	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))) ↔ ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
18	17	ralbii 3082	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
19	18	anbi2i 621	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
20	2, 19	bitri 274	1 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ∩ cin 3945   ⊆ wss 3946  𝒫 cpw 4606  ∪ cuni 4912   class class class wbr 5152  ωcom 7875   ≼ cdom 8971  Topctop 22878  1^stωc1stc 23424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-in 3953  df-ss 3963  df-pw 4608  df-uni 4913  df-1stc 23426

This theorem is referenced by:  1stcclb  23431  2ndc1stc  23438  1stcrest  23440  lly1stc  23483  tx1stc  23637  met1stc  24513