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Theorem iccpval 41869
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)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
Distinct variable group:   𝑖,𝑝,𝑀

Proof of Theorem iccpval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-iccp 41868 . . 3 RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21a1i 11 . 2 (𝑀 ∈ ℕ → RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 oveq2 6800 . . . . 5 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
43oveq2d 6808 . . . 4 (𝑚 = 𝑀 → (ℝ*𝑚 (0...𝑚)) = (ℝ*𝑚 (0...𝑀)))
5 oveq2 6800 . . . . 5 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
65raleqdv 3292 . . . 4 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
74, 6rabeqbidv 3344 . . 3 (𝑚 = 𝑀 → {𝑝 ∈ (ℝ*𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
87adantl 467 . 2 ((𝑀 ∈ ℕ ∧ 𝑚 = 𝑀) → {𝑝 ∈ (ℝ*𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
9 id 22 . 2 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ)
10 ovex 6822 . . . 4 (ℝ*𝑚 (0...𝑀)) ∈ V
1110rabex 4943 . . 3 {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
1211a1i 11 . 2 (𝑀 ∈ ℕ → {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ∈ V)
132, 8, 9, 12fvmptd 6430 1 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349   class class class wbr 4784  cmpt 4861  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  0cc0 10137  1c1 10138   + caddc 10140  *cxr 10274   < clt 10275  cn 11221  ...cfz 12532  ..^cfzo 12672  RePartciccp 41867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-iccp 41868
This theorem is referenced by:  iccpart  41870
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