Theorem crefi 34215

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 1154	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2		simp2 1155	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ⊆ 𝐽)
3	1, 2	sselpwd 5300	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4		crefi.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	iscref 34212	. . . 4 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
6	5	simprbi 502	. . 3 ⊢ (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
7	6	3ad2ant1 1151	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8		simp3 1156	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝑋 = ∪ 𝐶)
9		unieq 4884	. . . . 5 ⊢ (𝑦 = 𝐶 → ∪ 𝑦 = ∪ 𝐶)
10	9	eqeq2d 2774	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐶))
11		breq2 5114	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑧Ref𝑦 ↔ 𝑧Ref𝐶))
12	11	rexbidv 3189	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶))
13	10, 12	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
14	13	rspcv 3578	. 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 67	1 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4563  ∪ cuni 4873   class class class wbr 5110  Topctop 23031  Refcref 23640  CovHasRefccref 34210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-cref 34211

This theorem is referenced by:  crefdf  34216