Theorem etransclem12 42538

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 7166	. . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
3	2	oveq1d 7173	. . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀)))
4		eqeq2 2835	. . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
5	3, 4	rabeqbidv 3487	. 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 7191	. . . 4 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ V
8	7	rabex 5237	. . 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 6793	1 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3144  Vcvv 3496   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158   ↑_m cmap 8408  0cc0 10539  ℕ₀cn0 11900  ...cfz 12895  Σcsu 15044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161

This theorem is referenced by:  etransclem16  42542  etransclem24  42550  etransclem26  42552  etransclem28  42554  etransclem31  42557  etransclem32  42558  etransclem34  42560  etransclem35  42561  etransclem37  42563  etransclem38  42564