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Theorem dya2iocrfn 33009
Description: The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0 𝐽 = (topGen‘ran (,))
dya2ioc.1 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
dya2ioc.2 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
Assertion
Ref Expression
dya2iocrfn 𝑅 Fn (ran 𝐼 × ran 𝐼)
Distinct variable groups:   𝑥,𝑛   𝑥,𝐼   𝑣,𝑢,𝐼
Allowed substitution hints:   𝑅(𝑥,𝑣,𝑢,𝑛)   𝐼(𝑛)   𝐽(𝑥,𝑣,𝑢,𝑛)

Proof of Theorem dya2iocrfn
StepHypRef Expression
1 dya2ioc.2 . 2 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
2 vex 3470 . . 3 𝑢 ∈ V
3 vex 3470 . . 3 𝑣 ∈ V
42, 3xpex 7714 . 2 (𝑢 × 𝑣) ∈ V
51, 4fnmpoi 8029 1 𝑅 Fn (ran 𝐼 × ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   × cxp 5658  ran crn 5661   Fn wfn 6518  cfv 6523  (class class class)co 7384  cmpo 7386  1c1 11083   + caddc 11085   / cdiv 11843  2c2 12239  cz 12530  (,)cioo 13296  [,)cico 13298  cexp 13999  topGenctg 17355
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pow 5347  ax-pr 5411  ax-un 7699
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-oprab 7388  df-mpo 7389  df-1st 7948  df-2nd 7949
This theorem is referenced by:  dya2iocuni  33013
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