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Theorem etransclem12 39139
 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 . . 3 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})
21a1i 11 . 2 (𝜑𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛}))
3 oveq2 6557 . . . . 5 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43oveq1d 6564 . . . 4 (𝑛 = 𝑁 → ((0...𝑛) ↑𝑚 (0...𝑀)) = ((0...𝑁) ↑𝑚 (0...𝑀)))
5 eqeq2 2621 . . . 4 (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁))
64, 5rabeqbidv 3168 . . 3 (𝑛 = 𝑁 → {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
76adantl 481 . 2 ((𝜑𝑛 = 𝑁) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁})
8 etransclem12.2 . 2 (𝜑𝑁 ∈ ℕ0)
9 ovex 6577 . . . 4 ((0...𝑁) ↑𝑚 (0...𝑀)) ∈ V
109rabex 4740 . . 3 {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} ∈ V
1110a1i 11 . 2 (𝜑 → {𝑐 ∈ ((0...𝑁) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁} ∈ V)
122, 7, 8, 11fvmptd 6197 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 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  0cc0 9815  ℕ0cn0 11169  ...cfz 12197  Σcsu 14264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  etransclem16  39143  etransclem24  39151  etransclem26  39153  etransclem28  39155  etransclem31  39158  etransclem32  39159  etransclem34  39161  etransclem35  39162  etransclem37  39164  etransclem38  39165
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