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Theorem crefi 29242
 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 1054 . . . 4 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2 simp2 1055 . . . 4 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶𝐽)
31, 2ssexd 4733 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ V)
4 elpwg 4116 . . . 4 (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝐽𝐶𝐽))
54biimpar 501 . . 3 ((𝐶 ∈ V ∧ 𝐶𝐽) → 𝐶 ∈ 𝒫 𝐽)
63, 2, 5syl2anc 691 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ 𝒫 𝐽)
7 crefi.x . . . . 5 𝑋 = 𝐽
87iscref 29239 . . . 4 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
98simprbi 479 . . 3 (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
1093ad2ant1 1075 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
11 simp3 1056 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝑋 = 𝐶)
12 unieq 4380 . . . . 5 (𝑦 = 𝐶 𝑦 = 𝐶)
1312eqeq2d 2620 . . . 4 (𝑦 = 𝐶 → (𝑋 = 𝑦𝑋 = 𝐶))
14 breq2 4587 . . . . 5 (𝑦 = 𝐶 → (𝑧Ref𝑦𝑧Ref𝐶))
1514rexbidv 3034 . . . 4 (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶))
1613, 15imbi12d 333 . . 3 (𝑦 = 𝐶 → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
1716rspcv 3278 . 2 (𝐶 ∈ 𝒫 𝐽 → (∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) → (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
186, 10, 11, 17syl3c 64 1 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  Topctop 20517  Refcref 21115  CovHasRefccref 29237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-cref 29238 This theorem is referenced by:  crefdf  29243
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