Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dya2iocrfn Structured version   Visualization version   GIF version

Theorem dya2iocrfn 33955
Description: The function returning dyadic square covering for a given size has domain (ran 𝐼 Γ— ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0 𝐽 = (topGenβ€˜ran (,))
dya2ioc.1 𝐼 = (π‘₯ ∈ β„€, 𝑛 ∈ β„€ ↦ ((π‘₯ / (2↑𝑛))[,)((π‘₯ + 1) / (2↑𝑛))))
dya2ioc.2 𝑅 = (𝑒 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑒 Γ— 𝑣))
Assertion
Ref Expression
dya2iocrfn 𝑅 Fn (ran 𝐼 Γ— ran 𝐼)
Distinct variable groups:   π‘₯,𝑛   π‘₯,𝐼   𝑣,𝑒,𝐼
Allowed substitution hints:   𝑅(π‘₯,𝑣,𝑒,𝑛)   𝐼(𝑛)   𝐽(π‘₯,𝑣,𝑒,𝑛)

Proof of Theorem dya2iocrfn
StepHypRef Expression
1 dya2ioc.2 . 2 𝑅 = (𝑒 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑒 Γ— 𝑣))
2 vex 3467 . . 3 𝑒 ∈ V
3 vex 3467 . . 3 𝑣 ∈ V
42, 3xpex 7752 . 2 (𝑒 Γ— 𝑣) ∈ V
51, 4fnmpoi 8070 1 𝑅 Fn (ran 𝐼 Γ— ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   Γ— cxp 5670  ran crn 5673   Fn wfn 6537  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417  1c1 11137   + caddc 11139   / cdiv 11899  2c2 12295  β„€cz 12586  (,)cioo 13354  [,)cico 13356  β†‘cexp 14056  topGenctg 17416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990
This theorem is referenced by:  dya2iocuni  33959
  Copyright terms: Public domain W3C validator