Theorem dya2iocrfn 30341

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 3203	. . 3 ⊢ 𝑢 ∈ V
3		vex 3203	. . 3 ⊢ 𝑣 ∈ V
4	2, 3	xpex 6962	. 2 ⊢ (𝑢 × 𝑣) ∈ V
5	1, 4	fnmpt2i 7239	1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼)

Syntax hints:   = wceq 1483   × cxp 5112  ran crn 5115   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1c1 9937   + caddc 9939   / cdiv 10684  2c2 11070  ℤcz 11377  (,)cioo 12175  [,)cico 12177  ↑cexp 12860  topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  dya2iocuni  30345