Theorem etransclem12 46227

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 7357	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3	2	oveq1d 7364	. . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4		eqeq2 2741	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
5	3, 4	rabeqbidv 3413	. 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 7382	. . . 4 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ V
8	7	rabex 5278	. . 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 6953	1 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  0cc0 11009  ℕ₀cn0 12384  ...cfz 13410  Σcsu 15593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352

This theorem is referenced by:  etransclem16  46231  etransclem24  46239  etransclem26  46241  etransclem28  46243  etransclem31  46246  etransclem32  46247  etransclem34  46249  etransclem35  46250  etransclem37  46252  etransclem38  46253