Theorem itcoval 48397

Description: The value of the function that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by AV, 2-May-2024.)

Assertion

Ref	Expression
itcoval	⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))

Distinct variable group:   𝑔,𝐹,𝑖,𝑗

Allowed substitution hints:   𝑉(𝑔,𝑖,𝑗)

Proof of Theorem itcoval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-itco 48395	. 2 ⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
2		eqidd 2741	. . 3 ⊢ (𝑓 = 𝐹 → 0 = 0)
3		coeq1 5882	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
4	3	mpoeq3dv 7531	. . 3 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5		dmeq 5928	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	reseq2d 6011	. . . . 5 ⊢ (𝑓 = 𝐹 → ( I ↾ dom 𝑓) = ( I ↾ dom 𝐹))
7		id 22	. . . . 5 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
8	6, 7	ifeq12d 4569	. . . 4 ⊢ (𝑓 = 𝐹 → if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))
9	8	mpteq2dv 5268	. . 3 ⊢ (𝑓 = 𝐹 → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))
10	2, 4, 9	seqeq123d 14063	. 2 ⊢ (𝑓 = 𝐹 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
11		elex 3509	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		seqex 14056	. . 3 ⊢ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) ∈ V
13	12	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑉 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) ∈ V)
14	1, 10, 11, 13	fvmptd3 7054	1 ⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ifcif 4548   ↦ cmpt 5249   I cid 5592  dom cdm 5700   ↾ cres 5702   ∘ ccom 5704  ‘cfv 6575   ∈ cmpo 7452  0cc0 11186  ℕ₀cn0 12555  seqcseq 14054  IterCompcitco 48393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772  ax-inf2 9712

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-seq 14055  df-itco 48395

This theorem is referenced by:  itcoval0  48398  itcoval1  48399  itcoval2  48400  itcoval3  48401  itcovalsuc  48403