Theorem crefi 33846

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 1137	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2		simp2 1138	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ⊆ 𝐽)
3	1, 2	sselpwd 5328	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4		crefi.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	iscref 33843	. . . 4 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
6	5	simprbi 496	. . 3 ⊢ (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
7	6	3ad2ant1 1134	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8		simp3 1139	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝑋 = ∪ 𝐶)
9		unieq 4918	. . . . 5 ⊢ (𝑦 = 𝐶 → ∪ 𝑦 = ∪ 𝐶)
10	9	eqeq2d 2748	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐶))
11		breq2 5147	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑧Ref𝑦 ↔ 𝑧Ref𝐶))
12	11	rexbidv 3179	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶))
13	10, 12	imbi12d 344	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
14	13	rspcv 3618	. 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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143  Topctop 22899  Refcref 23510  CovHasRefccref 33841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-cref 33842

This theorem is referenced by:  crefdf  33847