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Theorem iscref 33843
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 𝐽𝐴)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.
StepHypRef Expression
1 pweq 4614 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2 unieq 4918 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
3 iscref.x . . . . . 6 𝑋 = 𝐽
42, 3eqtr4di 2795 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
54eqeq1d 2739 . . . 4 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
61ineq1d 4219 . . . . 5 (𝑗 = 𝐽 → (𝒫 𝑗𝐴) = (𝒫 𝐽𝐴))
76rexeqdv 3327 . . . 4 (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
85, 7imbi12d 344 . . 3 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
91, 8raleqbidv 3346 . 2 (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
10 df-cref 33842 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
119, 10elrab2 3695 1 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  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-in 3958  df-ss 3968  df-pw 4602  df-uni 4908  df-cref 33842
This theorem is referenced by:  creftop  33845  crefi  33846  crefss  33848  cmpcref  33849  cmppcmp  33857  dispcmp  33858  pcmplfin  33859
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