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Theorem iscref 31119
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 4528 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2 unieq 4822 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
3 iscref.x . . . . . 6 𝑋 = 𝐽
42, 3syl6eqr 2874 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
54eqeq1d 2823 . . . 4 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
61ineq1d 4163 . . . . 5 (𝑗 = 𝐽 → (𝒫 𝑗𝐴) = (𝒫 𝐽𝐴))
76rexeqdv 3397 . . . 4 (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
85, 7imbi12d 348 . . 3 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
91, 8raleqbidv 3386 . 2 (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
10 df-cref 31118 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
119, 10elrab2 3660 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 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wrex 3127  cin 3909  𝒫 cpw 4512   cuni 4811   class class class wbr 5039  Topctop 21477  Refcref 22086  CovHasRefccref 31117
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
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 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-in 3917  df-ss 3927  df-pw 4514  df-uni 4812  df-cref 31118
This theorem is referenced by:  creftop  31121  crefi  31122  crefss  31124  cmpcref  31125  cmppcmp  31133  dispcmp  31134  pcmplfin  31135
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