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Theorem iccpval 46073
Description: Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)
Assertion
Ref Expression
iccpval (𝑀 ∈ β„• β†’ (RePartβ€˜π‘€) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
Distinct variable group:   𝑖,𝑝,𝑀

Proof of Theorem iccpval
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 oveq2 7416 . . . 4 (π‘š = 𝑀 β†’ (0...π‘š) = (0...𝑀))
21oveq2d 7424 . . 3 (π‘š = 𝑀 β†’ (ℝ* ↑m (0...π‘š)) = (ℝ* ↑m (0...𝑀)))
3 oveq2 7416 . . . 4 (π‘š = 𝑀 β†’ (0..^π‘š) = (0..^𝑀))
43raleqdv 3325 . . 3 (π‘š = 𝑀 β†’ (βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)) ↔ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))))
52, 4rabeqbidv 3449 . 2 (π‘š = 𝑀 β†’ {𝑝 ∈ (ℝ* ↑m (0...π‘š)) ∣ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))} = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
6 df-iccp 46072 . 2 RePart = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ* ↑m (0...π‘š)) ∣ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
7 ovex 7441 . . 3 (ℝ* ↑m (0...𝑀)) ∈ V
87rabex 5332 . 2 {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))} ∈ V
95, 6, 8fvmpt 6998 1 (𝑀 ∈ β„• β†’ (RePartβ€˜π‘€) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  0cc0 11109  1c1 11110   + caddc 11112  β„*cxr 11246   < clt 11247  β„•cn 12211  ...cfz 13483  ..^cfzo 13626  RePartciccp 46071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-iccp 46072
This theorem is referenced by:  iccpart  46074
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