Theorem dya2iocrfn 34436

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 3444	. . 3 ⊢ 𝑢 ∈ V
3		vex 3444	. . 3 ⊢ 𝑣 ∈ V
4	2, 3	xpex 7698	. 2 ⊢ (𝑢 × 𝑣) ∈ V
5	1, 4	fnmpoi 8014	1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼)

Syntax hints:   = wceq 1541   × cxp 5622  ran crn 5625   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1c1 11027   + caddc 11029   / cdiv 11794  2c2 12200  ℤcz 12488  (,)cioo 13261  [,)cico 13263  ↑cexp 13984  topGenctg 17357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934

This theorem is referenced by:  dya2iocuni  34440