Theorem iscref 34021

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 4570	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2		unieq 4876	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		iscref.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	4	eqeq1d 2739	. . . 4 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	1	ineq1d 4173	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝐽 ∩ 𝐴))
7	6	rexeqdv 3299	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8	5, 7	imbi12d 344	. . 3 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
9	1, 8	raleqbidv 3318	. 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 34020	. 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
11	9, 10	elrab2 3651	1 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100  Topctop 22849  Refcref 23458  CovHasRefccref 34019

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-pw 4558  df-uni 4866  df-cref 34020

This theorem is referenced by:  creftop  34023  crefi  34024  crefss  34026  cmpcref  34027  cmppcmp  34035  dispcmp  34036  pcmplfin  34037