Theorem itcoval 45075

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 45073	. 2 ⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
2		eqidd 2799	. . 3 ⊢ (𝑓 = 𝐹 → 0 = 0)
3		coeq1 5692	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
4	3	mpoeq3dv 7212	. . 3 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5		dmeq 5736	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	reseq2d 5818	. . . . 5 ⊢ (𝑓 = 𝐹 → ( I ↾ dom 𝑓) = ( I ↾ dom 𝐹))
7		id 22	. . . . 5 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
8	6, 7	ifeq12d 4445	. . . 4 ⊢ (𝑓 = 𝐹 → if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))
9	8	mpteq2dv 5126	. . 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 13373	. 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 3459	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		seqex 13366	. . 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 6768	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 1538   ∈ wcel 2111  Vcvv 3441  ifcif 4425   ↦ cmpt 5110   I cid 5424  dom cdm 5519   ↾ cres 5521   ∘ ccom 5523  ‘cfv 6324   ∈ cmpo 7137  0cc0 10526  ℕ₀cn0 11885  seqcseq 13364  IterCompcitco 45071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seq 13365  df-itco 45073

This theorem is referenced by:  itcoval0  45076  itcoval1  45077  itcoval2  45078  itcoval3  45079  itcovalsuc  45081