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Theorem crefi 29696
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 1059 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2 simp2 1060 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶𝐽)
31, 2sselpwd 4767 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4 crefi.x . . . . 5 𝑋 = 𝐽
54iscref 29693 . . . 4 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
65simprbi 480 . . 3 (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
763ad2ant1 1080 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
8 simp3 1061 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝑋 = 𝐶)
9 unieq 4410 . . . . 5 (𝑦 = 𝐶 𝑦 = 𝐶)
109eqeq2d 2631 . . . 4 (𝑦 = 𝐶 → (𝑋 = 𝑦𝑋 = 𝐶))
11 breq2 4617 . . . . 5 (𝑦 = 𝐶 → (𝑧Ref𝑦𝑧Ref𝐶))
1211rexbidv 3045 . . . 4 (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶))
1310, 12imbi12d 334 . . 3 (𝑦 = 𝐶 → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
1413rspcv 3291 . 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 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  Topctop 20617  Refcref 21215  CovHasRefccref 29691
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  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  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-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-cref 29692
This theorem is referenced by:  crefdf  29697
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