Theorem iccpval 44755

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 7263	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7271	. . 3 ⊢ (𝑚 = 𝑀 → (ℝ* ↑m (0...𝑚)) = (ℝ* ↑m (0...𝑀)))
3		oveq2 7263	. . . 4 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
4	3	raleqdv 3339	. . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
5	2, 4	rabeqbidv 3410	. 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 44754	. 2 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
7		ovex 7288	. . 3 ⊢ (ℝ* ↑m (0...𝑀)) ∈ V
8	7	rabex 5251	. 2 ⊢ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
9	5, 6, 8	fvmpt 6857	1 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  0cc0 10802  1c1 10803   + caddc 10805  ℝ^*cxr 10939   < clt 10940  ℕcn 11903  ...cfz 13168  ..^cfzo 13311  RePartciccp 44753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-iccp 44754

This theorem is referenced by:  iccpart  44756