Theorem iccpval 47407

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 7440	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7448	. . 3 ⊢ (𝑚 = 𝑀 → (ℝ* ↑m (0...𝑚)) = (ℝ* ↑m (0...𝑀)))
3		oveq2 7440	. . . 4 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
4	3	raleqdv 3325	. . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
5	2, 4	rabeqbidv 3454	. 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 47406	. 2 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
7		ovex 7465	. . 3 ⊢ (ℝ* ↑m (0...𝑀)) ∈ V
8	7	rabex 5338	. 2 ⊢ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
9	5, 6, 8	fvmpt 7015	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 2107  ∀wral 3060  {crab 3435   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  0cc0 11156  1c1 11157   + caddc 11159  ℝ^*cxr 11295   < clt 11296  ℕcn 12267  ...cfz 13548  ..^cfzo 13695  RePartciccp 47405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-iccp 47406

This theorem is referenced by:  iccpart  47408