Theorem crefi 31465

Description: The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Hypothesis

Ref	Expression
crefi.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
crefi	⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑧,𝐶

Allowed substitution hint:   𝑋(𝑧)

Proof of Theorem crefi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1138	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2		simp2 1139	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ⊆ 𝐽)
3	1, 2	sselpwd 5204	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4		crefi.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	iscref 31462	. . . 4 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
6	5	simprbi 500	. . 3 ⊢ (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
7	6	3ad2ant1 1135	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8		simp3 1140	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝑋 = ∪ 𝐶)
9		unieq 4816	. . . . 5 ⊢ (𝑦 = 𝐶 → ∪ 𝑦 = ∪ 𝐶)
10	9	eqeq2d 2747	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐶))
11		breq2 5043	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑧Ref𝑦 ↔ 𝑧Ref𝐶))
12	11	rexbidv 3206	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶))
13	10, 12	imbi12d 348	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
14	13	rspcv 3522	. 2 ⊢ (𝐶 ∈ 𝒫 𝐽 → (∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) → (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
15	3, 7, 8, 14	syl3c 66	1 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052   ∩ cin 3852   ⊆ wss 3853  𝒫 cpw 4499  ∪ cuni 4805   class class class wbr 5039  Topctop 21744  Refcref 22353  CovHasRefccref 31460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-cref 31461

This theorem is referenced by:  crefdf  31466