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Theorem iscref 30509
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 4381 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2 unieq 4679 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
3 iscref.x . . . . . 6 𝑋 = 𝐽
42, 3syl6eqr 2831 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
54eqeq1d 2779 . . . 4 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
61ineq1d 4035 . . . . 5 (𝑗 = 𝐽 → (𝒫 𝑗𝐴) = (𝒫 𝐽𝐴))
76rexeqdv 3340 . . . 4 (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
85, 7imbi12d 336 . . 3 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
91, 8raleqbidv 3325 . 2 (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
10 df-cref 30508 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
119, 10elrab2 3575 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 198  wa 386   = wceq 1601  wcel 2106  wral 3089  wrex 3090  cin 3790  𝒫 cpw 4378   cuni 4671   class class class wbr 4886  Topctop 21105  Refcref 21714  CovHasRefccref 30507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-in 3798  df-ss 3805  df-pw 4380  df-uni 4672  df-cref 30508
This theorem is referenced by:  creftop  30511  crefi  30512  crefss  30514  cmpcref  30515  cmppcmp  30523  dispcmp  30524  pcmplfin  30525
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