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Theorem dya2iocrfn 31423
 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 3503 . . 3 𝑢 ∈ V
3 vex 3503 . . 3 𝑣 ∈ V
42, 3xpex 7465 . 2 (𝑢 × 𝑣) ∈ V
51, 4fnmpoi 7759 1 𝑅 Fn (ran 𝐼 × ran 𝐼)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   × cxp 5552  ran crn 5555   Fn wfn 6347  ‘cfv 6352  (class class class)co 7148   ∈ cmpo 7150  1c1 10527   + caddc 10529   / cdiv 11286  2c2 11681  ℤcz 11970  (,)cioo 12728  [,)cico 12730  ↑cexp 13419  topGenctg 16701 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681 This theorem is referenced by:  dya2iocuni  31427
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