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Theorem etransclem12 45634
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 7428 . . . 4 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
32oveq1d 7435 . . 3 (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4 eqeq2 2740 . . 3 (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁))
53, 4rabeqbidv 3446 . 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 7453 . . . 4 ((0...𝑁) ↑m (0...𝑀)) ∈ V
87rabex 5334 . . 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 7028 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 1534  wcel 2099  {crab 3429  Vcvv 3471  cmpt 5231  cfv 6548  (class class class)co 7420  m cmap 8844  0cc0 11138  0cn0 12502  ...cfz 13516  Σcsu 15664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423
This theorem is referenced by:  etransclem16  45638  etransclem24  45646  etransclem26  45648  etransclem28  45650  etransclem31  45653  etransclem32  45654  etransclem34  45656  etransclem35  45657  etransclem37  45659  etransclem38  45660
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