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Theorem itcoval 45005
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 (𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (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.
StepHypRef Expression
1 df-itco 45003 . 2 IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
2 eqidd 2825 . . 3 (𝑓 = 𝐹 → 0 = 0)
3 coeq1 5715 . . . 4 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
43mpoeq3dv 7226 . . 3 (𝑓 = 𝐹 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)))
5 dmeq 5759 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65reseq2d 5840 . . . . 5 (𝑓 = 𝐹 → ( I ↾ dom 𝑓) = ( I ↾ dom 𝐹))
7 id 22 . . . . 5 (𝑓 = 𝐹𝑓 = 𝐹)
86, 7ifeq12d 4470 . . . 4 (𝑓 = 𝐹 → if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))
98mpteq2dv 5148 . . 3 (𝑓 = 𝐹 → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))
102, 4, 9seqeq123d 13382 . 2 (𝑓 = 𝐹 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
11 elex 3498 . 2 (𝐹𝑉𝐹 ∈ V)
12 seqex 13375 . . 3 seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) ∈ V
1312a1i 11 . 2 (𝐹𝑉 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) ∈ V)
141, 10, 11, 13fvmptd3 6782 1 (𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  ifcif 4450  cmpt 5132   I cid 5446  dom cdm 5542  cres 5544  ccom 5546  cfv 6343  cmpo 7151  0cc0 10535  0cn0 11894  seqcseq 13373  IterCompcitco 45001
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seq 13374  df-itco 45003
This theorem is referenced by:  itcoval0  45006  itcoval1  45007  itcoval2  45008  itcoval3  45009  itcovalsuc  45011
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