Theorem 1stcclb 21247

Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Hypothesis

Ref	Expression
1stcclb.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
1stcclb	⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stcclb.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	is1stc2 21245	. . 3 ⊢ (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
3	2	simprbi 480	. 2 ⊢ (𝐽 ∈ 1st𝜔 → ∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
4		eleq1 2689	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
5		eleq1 2689	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑧 ↔ 𝐴 ∈ 𝑧))
6	5	anbi1d 741	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
7	6	rexbidv 3052	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
8	4, 7	imbi12d 334	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
9	8	ralbidv 2986	. . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
10	9	anbi2d 740	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
11	10	rexbidv 3052	. . 3 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
12	11	rspcv 3305	. 2 ⊢ (𝐴 ∈ 𝑋 → (∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
13	3, 12	mpan9 486	1 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  ωcom 7065   ≼ cdom 7953  Topctop 20698  1^st𝜔c1stc 21240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-1stc 21242

This theorem is referenced by:  1stcfb  21248  1stcrest  21256  lly1stc  21299  tx1stc  21453