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Theorem iscref 33578
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 4618 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2 unieq 4920 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
3 iscref.x . . . . . 6 𝑋 = 𝐽
42, 3eqtr4di 2783 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
54eqeq1d 2727 . . . 4 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
61ineq1d 4209 . . . . 5 (𝑗 = 𝐽 → (𝒫 𝑗𝐴) = (𝒫 𝐽𝐴))
76rexeqdv 3315 . . . 4 (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
85, 7imbi12d 343 . . 3 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
91, 8raleqbidv 3329 . 2 (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
10 df-cref 33577 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
119, 10elrab2 3682 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 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  cin 3943  𝒫 cpw 4604   cuni 4909   class class class wbr 5149  Topctop 22844  Refcref 23455  CovHasRefccref 33576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961  df-pw 4606  df-uni 4910  df-cref 33577
This theorem is referenced by:  creftop  33580  crefi  33581  crefss  33583  cmpcref  33584  cmppcmp  33592  dispcmp  33593  pcmplfin  33594
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