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Theorem iccpval 45697
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 7369 . . . 4 (π‘š = 𝑀 β†’ (0...π‘š) = (0...𝑀))
21oveq2d 7377 . . 3 (π‘š = 𝑀 β†’ (ℝ* ↑m (0...π‘š)) = (ℝ* ↑m (0...𝑀)))
3 oveq2 7369 . . . 4 (π‘š = 𝑀 β†’ (0..^π‘š) = (0..^𝑀))
43raleqdv 3312 . . 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 3423 . 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 45696 . 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 7394 . . 3 (ℝ* ↑m (0...𝑀)) ∈ V
87rabex 5293 . 2 {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ βˆ€π‘– ∈ (0..^𝑀)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1))} ∈ V
95, 6, 8fvmpt 6952 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 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  0cc0 11059  1c1 11060   + caddc 11062  β„*cxr 11196   < clt 11197  β„•cn 12161  ...cfz 13433  ..^cfzo 13576  RePartciccp 45695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-iccp 45696
This theorem is referenced by:  iccpart  45698
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