Theorem is1stc2 22052

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 22051	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3		elin 4171	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝒫 𝑧))
4		velpw 4546	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
5	4	anbi2i 624	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧))
6	3, 5	bitri 277	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧))
7	6	anbi2i 624	. . . . . . . . . . 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 277	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
10	9	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
11		eluni 4843	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12		df-rex 3146	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
13	10, 11, 12	3bitr4i 305	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
14	13	imbi2i 338	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
15	14	ralbii 3167	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
16	15	anbi2i 624	. . . . 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 3249	. . . 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 3167	. . 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 624	. 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 277	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 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840   class class class wbr 5068  ωcom 7582   ≼ cdom 8509  Topctop 21503  1^stωc1stc 22047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-in 3945  df-ss 3954  df-pw 4543  df-uni 4841  df-1stc 22049

This theorem is referenced by:  1stcclb  22054  2ndc1stc  22061  1stcrest  22063  lly1stc  22106  tx1stc  22260  met1stc  23133