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Theorem etransclem12 45696
Description: 𝐶 applied to 𝑁. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem12.1 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})
etransclem12.2 (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
etransclem12 (𝜑 → (𝐶𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
Distinct variable groups:   𝑀,𝑐,𝑛   𝑁,𝑐,𝑛   𝑗,𝑛   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑗,𝑐)   𝐶(𝑗,𝑛,𝑐)   𝑀(𝑗)   𝑁(𝑗)

Proof of Theorem etransclem12
StepHypRef Expression
1 etransclem12.1 . 2 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})
2 oveq2 7423 . . . 4 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32oveq1d 7430 . . 3 (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4 eqeq2 2737 . . 3 (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁))
53, 4rabeqbidv 3437 . 2 (𝑛 = 𝑁 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
6 etransclem12.2 . 2 (𝜑𝑁 ∈ ℕ0)
7 ovex 7448 . . . 4 ((0...𝑁) ↑m (0...𝑀)) ∈ V
87rabex 5329 . . 3 {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} ∈ V
98a1i 11 . 2 (𝜑 → {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} ∈ V)
101, 5, 6, 9fvmptd3 7022 1 (𝜑 → (𝐶𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3419  Vcvv 3463  cmpt 5226  cfv 6542  (class class class)co 7415  m cmap 8841  0cc0 11136  0cn0 12500  ...cfz 13514  Σcsu 15662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418
This theorem is referenced by:  etransclem16  45700  etransclem24  45708  etransclem26  45710  etransclem28  45712  etransclem31  45715  etransclem32  45716  etransclem34  45718  etransclem35  45719  etransclem37  45721  etransclem38  45722
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