Theorem iscref 31197

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
iscref.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscref	⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))

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

Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem iscref

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4513	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2		unieq 4811	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		iscref.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2851	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	4	eqeq1d 2800	. . . 4 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	1	ineq1d 4138	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝐽 ∩ 𝐴))
7	6	rexeqdv 3365	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8	5, 7	imbi12d 348	. . 3 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
9	1, 8	raleqbidv 3354	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
10		df-cref 31196	. 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
11	9, 10	elrab2 3631	1 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  Topctop 21498  Refcref 22107  CovHasRefccref 31195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-cref 31196

This theorem is referenced by:  creftop  31199  crefi  31200  crefss  31202  cmpcref  31203  cmppcmp  31211  dispcmp  31212  pcmplfin  31213