Theorem numclwwlkfvc 26604

Description: Value of function 𝐶, mapping a nonnegative number n to the closed walks having length n. (Contributed by Alexander van der Vekens, 14-Sep-2018.)

Hypothesis

Ref	Expression
numclwwlk.c	⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))

Assertion

Ref	Expression
numclwwlkfvc	⊢ (𝑁 ∈ ℕ0 → (𝐶‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))

Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉

Allowed substitution hint:   𝐶(𝑛)

Proof of Theorem numclwwlkfvc

Step	Hyp	Ref	Expression
1		fveq2 6103	. 2 ⊢ (𝑛 = 𝑁 → ((𝑉 ClWWalksN 𝐸)‘𝑛) = ((𝑉 ClWWalksN 𝐸)‘𝑁))
2		numclwwlk.c	. 2 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
3		fvex 6113	. 2 ⊢ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∈ V
4	1, 2, 3	fvmpt 6191	1 ⊢ (𝑁 ∈ ℕ0 → (𝐶‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℕ₀cn0 11169   ClWWalksN cclwwlkn 26277

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-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

This theorem is referenced by:  extwwlkfablem2  26605  numclwwlkun  26606  numclwwlkffin  26609  numclwwlkovfel2  26610  numclwwlkovf2  26611  numclwwlkovf2ex  26613  numclwwlkovgel  26615  extwwlkfab  26617  numclwwlkqhash  26627  numclwwlk2lem1  26629  numclwlk2lem2f  26630  numclwlk2lem2f1o  26632  numclwwlk3lem  26635  numclwwlk4  26637  numclwwlk7  26641