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Theorem etransclem12 40974
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 6800 . . . . 5 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43oveq1d 6807 . . . 4 (𝑛 = 𝑁 → ((0...𝑛) ↑𝑚 (0...𝑀)) = ((0...𝑁) ↑𝑚 (0...𝑀)))
5 eqeq2 2781 . . . 4 (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑁))
64, 5rabeqbidv 3344 . . 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 467 . 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 6822 . . . 4 ((0...𝑁) ↑𝑚 (0...𝑀)) ∈ V
109rabex 4943 . . 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 6430 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 1630  wcel 2144  {crab 3064  Vcvv 3349  cmpt 4861  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  0cc0 10137  0cn0 11493  ...cfz 12532  Σcsu 14623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795
This theorem is referenced by:  etransclem16  40978  etransclem24  40986  etransclem26  40988  etransclem28  40990  etransclem31  40993  etransclem32  40994  etransclem34  40996  etransclem35  40997  etransclem37  40999  etransclem38  41000
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