Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crefi Structured version   Visualization version   GIF version

Theorem crefi 33846
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 1137 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2 simp2 1138 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶𝐽)
31, 2sselpwd 5328 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4 crefi.x . . . . 5 𝑋 = 𝐽
54iscref 33843 . . . 4 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
65simprbi 496 . . 3 (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
763ad2ant1 1134 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
8 simp3 1139 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝑋 = 𝐶)
9 unieq 4918 . . . . 5 (𝑦 = 𝐶 𝑦 = 𝐶)
109eqeq2d 2748 . . . 4 (𝑦 = 𝐶 → (𝑋 = 𝑦𝑋 = 𝐶))
11 breq2 5147 . . . . 5 (𝑦 = 𝐶 → (𝑧Ref𝑦𝑧Ref𝐶))
1211rexbidv 3179 . . . 4 (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶))
1310, 12imbi12d 344 . . 3 (𝑦 = 𝐶 → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
1413rspcv 3618 . 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 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  Topctop 22899  Refcref 23510  CovHasRefccref 33841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-cref 33842
This theorem is referenced by:  crefdf  33847
  Copyright terms: Public domain W3C validator