Theorem itcoval 49243

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 49241	. 2 ⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
2		eqidd 2762	. . 3 ⊢ (𝑓 = 𝐹 → 0 = 0)
3		coeq1 5825	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
4	3	mpoeq3dv 7469	. . 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 5961	. . . . 5 ⊢ (𝑓 = 𝐹 → ( I ↾ dom 𝑓) = ( I ↾ dom 𝐹))
7		id 22	. . . . 5 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
8	6, 7	ifeq12d 4499	. . . 4 ⊢ (𝑓 = 𝐹 → if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))
9	8	mpteq2dv 5191	. . 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 14016	. 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 3474	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		seqex 14009	. . 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 6993	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 1559   ∈ wcel 2141  Vcvv 3453  ifcif 4477   ↦ cmpt 5178   I cid 5537  dom cdm 5643   ↾ cres 5645   ∘ ccom 5647  ‘cfv 6515   ∈ cmpo 7392  0cc0 11066  ℕ₀cn0 12474  seqcseq 14007  IterCompcitco 49239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712  ax-inf2 9589

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-seq 14008  df-itco 49241

This theorem is referenced by:  itcoval0  49244  itcoval1  49245  itcoval2  49246  itcoval3  49247  itcovalsuc  49249