Theorem itcoval 48583

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 48581	. 2 ⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
2		eqidd 2731	. . 3 ⊢ (𝑓 = 𝐹 → 0 = 0)
3		coeq1 5829	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
4	3	mpoeq3dv 7475	. . 3 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5		dmeq 5875	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	reseq2d 5958	. . . . 5 ⊢ (𝑓 = 𝐹 → ( I ↾ dom 𝑓) = ( I ↾ dom 𝐹))
7		id 22	. . . . 5 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
8	6, 7	ifeq12d 4518	. . . 4 ⊢ (𝑓 = 𝐹 → if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))
9	8	mpteq2dv 5209	. . 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 13985	. 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 3476	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		seqex 13978	. . 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 6998	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 1540   ∈ wcel 2109  Vcvv 3455  ifcif 4496   ↦ cmpt 5196   I cid 5540  dom cdm 5646   ↾ cres 5648   ∘ ccom 5650  ‘cfv 6519   ∈ cmpo 7396  0cc0 11086  ℕ₀cn0 12458  seqcseq 13976  IterCompcitco 48579

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 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718  ax-inf2 9612

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-seq 13977  df-itco 48581

This theorem is referenced by:  itcoval0  48584  itcoval1  48585  itcoval2  48586  itcoval3  48587  itcovalsuc  48589