Theorem etransclem12 47018

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)(𝑐‘𝑗) = 𝑛})
etransclem12.2	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
etransclem12	⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 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

Step	Hyp	Ref	Expression
1		etransclem12.1	. 2 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
2		oveq2 7427	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3	2	oveq1d 7434	. . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4		eqeq2 2777	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
5	3, 4	rabeqbidv 3436	. 2 ⊢ (𝑛 = 𝑁 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
6		etransclem12.2	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
7		ovex 7452	. . . 4 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ V
8	7	rabex 5311	. . 3 ⊢ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ V)
10	1, 5, 6, 9	fvmptd3 7017	1 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  0cc0 11115  ℕ₀cn0 12519  ...cfz 13551  Σcsu 15761

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422

This theorem is used by:  etransclem16  47022  etransclem24  47030  etransclem26  47032  etransclem28  47034  etransclem31  47037  etransclem32  47038  etransclem34  47040  etransclem35  47041  etransclem37  47043  etransclem38  47044