Theorem iccpval 47446

Description: Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)

Assertion

Ref	Expression
iccpval	⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})

Distinct variable group:   𝑖,𝑝,𝑀

Proof of Theorem iccpval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7349	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7357	. . 3 ⊢ (𝑚 = 𝑀 → (ℝ* ↑m (0...𝑚)) = (ℝ* ↑m (0...𝑀)))
3		oveq2 7349	. . . 4 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
4	3	raleqdv 3292	. . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
5	2, 4	rabeqbidv 3413	. 2 ⊢ (𝑚 = 𝑀 → {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
6		df-iccp 47445	. 2 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
7		ovex 7374	. . 3 ⊢ (ℝ* ↑m (0...𝑀)) ∈ V
8	7	rabex 5272	. 2 ⊢ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
9	5, 6, 8	fvmpt 6924	1 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745  0cc0 11001  1c1 11002   + caddc 11004  ℝ^*cxr 11140   < clt 11141  ℕcn 12120  ...cfz 13402  ..^cfzo 13549  RePartciccp 47444

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-iccp 47445

This theorem is referenced by:  iccpart  47447