Theorem is1stc2 23475

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 23474	. 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))))
3		elin 3915	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑦 ∩ 𝒫 𝑧) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝒫 𝑧))
4		velpw 4554	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
5	4	anbi2i 631	. . . . . . . . . . . . 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 631	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ (𝑥 ∈ 𝑤 ∧ (𝑤 ∈ 𝑦 ∧ 𝑤 ⊆ 𝑧)))
8		an12 653	. . . . . . . . . . 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 1862	. . . . . . . . 9 ⊢ (∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
11		eluni 4862	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ (𝑦 ∩ 𝒫 𝑧)))
12		df-rex 3081	. . . . . . . . 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 3102	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
16	15	anbi2i 631	. . . . 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 3103	. . . 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 3102	. . 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 631	. 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 1554  ∃wex 1793   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4549  ∪ cuni 4859   class class class wbr 5094  ωcom 7835   ≼ cdom 8914  Topctop 22926  1^stωc1stc 23470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-in 3906  df-ss 3916  df-pw 4551  df-uni 4860  df-1stc 23472

This theorem is referenced by:  1stcclb  23477  2ndc1stc  23484  1stcrest  23486  lly1stc  23529  tx1stc  23683  met1stc  24554