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Theorem iscref 29690
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 4133 . . 3 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2 unieq 4410 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
3 iscref.x . . . . . 6 𝑋 = 𝐽
42, 3syl6eqr 2673 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
54eqeq1d 2623 . . . 4 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
61ineq1d 3791 . . . . 5 (𝑗 = 𝐽 → (𝒫 𝑗𝐴) = (𝒫 𝐽𝐴))
76rexeqdv 3134 . . . 4 (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
85, 7imbi12d 334 . . 3 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
91, 8raleqbidv 3141 . 2 (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
10 df-cref 29689 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
119, 10elrab2 3348 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  Topctop 20617  Refcref 21215  CovHasRefccref 29688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-cref 29689
This theorem is referenced by:  creftop  29692  crefi  29693  crefss  29695  cmpcref  29696  cmppcmp  29704  dispcmp  29705  pcmplfin  29706
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