Theorem dya2iocrfn 34456

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

Step	Hyp	Ref	Expression
1		dya2ioc.2	. 2 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
2		vex 3446	. . 3 ⊢ 𝑢 ∈ V
3		vex 3446	. . 3 ⊢ 𝑣 ∈ V
4	2, 3	xpex 7708	. 2 ⊢ (𝑢 × 𝑣) ∈ V
5	1, 4	fnmpoi 8024	1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼)

Syntax hints:   = wceq 1542   × cxp 5630  ran crn 5633   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1c1 11039   + caddc 11041   / cdiv 11806  2c2 12212  ℤcz 12500  (,)cioo 13273  [,)cico 13275  ↑cexp 13996  topGenctg 17369

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944

This theorem is referenced by:  dya2iocuni  34460