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Theorem crefi 31010
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
crefi.x 𝑋 = 𝐽
Assertion
Ref Expression
crefi ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑧,𝐶
Allowed substitution hint:   𝑋(𝑧)

Proof of Theorem crefi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1128 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2 simp2 1129 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶𝐽)
31, 2sselpwd 5221 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4 crefi.x . . . . 5 𝑋 = 𝐽
54iscref 31007 . . . 4 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
65simprbi 497 . . 3 (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
763ad2ant1 1125 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
8 simp3 1130 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝑋 = 𝐶)
9 unieq 4838 . . . . 5 (𝑦 = 𝐶 𝑦 = 𝐶)
109eqeq2d 2829 . . . 4 (𝑦 = 𝐶 → (𝑋 = 𝑦𝑋 = 𝐶))
11 breq2 5061 . . . . 5 (𝑦 = 𝐶 → (𝑧Ref𝑦𝑧Ref𝐶))
1211rexbidv 3294 . . . 4 (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶))
1310, 12imbi12d 346 . . 3 (𝑦 = 𝐶 → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
1413rspcv 3615 . 2 (𝐶 ∈ 𝒫 𝐽 → (∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) → (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
153, 7, 8, 14syl3c 66 1 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  Topctop 21429  Refcref 22038  CovHasRefccref 31005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-cref 31006
This theorem is referenced by:  crefdf  31011
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