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

Theorem crefss 32817
Description: The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
crefss (𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Proof of Theorem crefss
Dummy variables 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sslin 4233 . . . . . . 7 (𝐴𝐵 → (𝒫 𝑗𝐴) ⊆ (𝒫 𝑗𝐵))
2 ssrexv 4050 . . . . . . 7 ((𝒫 𝑗𝐴) ⊆ (𝒫 𝑗𝐵) → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦))
31, 2syl 17 . . . . . 6 (𝐴𝐵 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦))
43imim2d 57 . . . . 5 (𝐴𝐵 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) → ( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
54ralimdv 3169 . . . 4 (𝐴𝐵 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) → ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
65anim2d 612 . . 3 (𝐴𝐵 → ((𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)) → (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦))))
7 eqid 2732 . . . 4 𝑗 = 𝑗
87iscref 32812 . . 3 (𝑗 ∈ CovHasRef𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)))
97iscref 32812 . . 3 (𝑗 ∈ CovHasRef𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
106, 8, 93imtr4g 295 . 2 (𝐴𝐵 → (𝑗 ∈ CovHasRef𝐴𝑗 ∈ CovHasRef𝐵))
1110ssrdv 3987 1 (𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   class class class wbr 5147  Topctop 22386  Refcref 22997  CovHasRefccref 32810
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-in 3954  df-ss 3964  df-pw 4603  df-uni 4908  df-cref 32811
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator