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Theorem dya2iocrfn 33808
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 3472 . . 3 𝑒 ∈ V
3 vex 3472 . . 3 𝑣 ∈ V
42, 3xpex 7737 . 2 (𝑒 Γ— 𝑣) ∈ V
51, 4fnmpoi 8055 1 𝑅 Fn (ran 𝐼 Γ— ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   Γ— cxp 5667  ran crn 5670   Fn wfn 6532  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1c1 11113   + caddc 11115   / cdiv 11875  2c2 12271  β„€cz 12562  (,)cioo 13330  [,)cico 13332  β†‘cexp 14032  topGenctg 17392
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  dya2iocuni  33812
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