Theorem crefi 33123

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 1134	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2		simp2 1135	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ⊆ 𝐽)
3	1, 2	sselpwd 5327	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4		crefi.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	iscref 33120	. . . 4 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
6	5	simprbi 495	. . 3 ⊢ (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
7	6	3ad2ant1 1131	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8		simp3 1136	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝑋 = ∪ 𝐶)
9		unieq 4920	. . . . 5 ⊢ (𝑦 = 𝐶 → ∪ 𝑦 = ∪ 𝐶)
10	9	eqeq2d 2741	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐶))
11		breq2 5153	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑧Ref𝑦 ↔ 𝑧Ref𝐶))
12	11	rexbidv 3176	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶))
13	10, 12	imbi12d 343	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
14	13	rspcv 3609	. 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 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  Topctop 22617  Refcref 23228  CovHasRefccref 33118

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-cref 33119

This theorem is referenced by:  crefdf  33124