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Theorem crefi 31200
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 1133 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2 simp2 1134 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶𝐽)
31, 2sselpwd 5197 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4 crefi.x . . . . 5 𝑋 = 𝐽
54iscref 31197 . . . 4 (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
65simprbi 500 . . 3 (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
763ad2ant1 1130 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦))
8 simp3 1135 . 2 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → 𝑋 = 𝐶)
9 unieq 4814 . . . . 5 (𝑦 = 𝐶 𝑦 = 𝐶)
109eqeq2d 2812 . . . 4 (𝑦 = 𝐶 → (𝑋 = 𝑦𝑋 = 𝐶))
11 breq2 5037 . . . . 5 (𝑦 = 𝐶 → (𝑧Ref𝑦𝑧Ref𝐶))
1211rexbidv 3259 . . . 4 (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶))
1310, 12imbi12d 348 . . 3 (𝑦 = 𝐶 → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦) ↔ (𝑋 = 𝐶 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)))
1413rspcv 3569 . 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 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  cin 3883  wss 3884  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  Topctop 21501  Refcref 22110  CovHasRefccref 31195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-cref 31196
This theorem is referenced by:  crefdf  31201
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