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Theorem itcoval 47812
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 β†’ (IterCompβ€˜πΉ) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ 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 47810 . 2 IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓))))
2 eqidd 2729 . . 3 (𝑓 = 𝐹 β†’ 0 = 0)
3 coeq1 5864 . . . 4 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
43mpoeq3dv 7505 . . 3 (𝑓 = 𝐹 β†’ (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5 dmeq 5910 . . . . . 6 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
65reseq2d 5989 . . . . 5 (𝑓 = 𝐹 β†’ ( I β†Ύ dom 𝑓) = ( I β†Ύ dom 𝐹))
7 id 22 . . . . 5 (𝑓 = 𝐹 β†’ 𝑓 = 𝐹)
86, 7ifeq12d 4553 . . . 4 (𝑓 = 𝐹 β†’ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))
98mpteq2dv 5254 . . 3 (𝑓 = 𝐹 β†’ (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓)) = (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹)))
102, 4, 9seqeq123d 14015 . 2 (𝑓 = 𝐹 β†’ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓))) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))))
11 elex 3492 . 2 (𝐹 ∈ 𝑉 β†’ 𝐹 ∈ V)
12 seqex 14008 . . 3 seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))) ∈ V
1312a1i 11 . 2 (𝐹 ∈ 𝑉 β†’ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))) ∈ V)
141, 10, 11, 13fvmptd3 7033 1 (𝐹 ∈ 𝑉 β†’ (IterCompβ€˜πΉ) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ifcif 4532   ↦ cmpt 5235   I cid 5579  dom cdm 5682   β†Ύ cres 5684   ∘ ccom 5686  β€˜cfv 6553   ∈ cmpo 7428  0cc0 11146  β„•0cn0 12510  seqcseq 14006  IterCompcitco 47808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-seq 14007  df-itco 47810
This theorem is referenced by:  itcoval0  47813  itcoval1  47814  itcoval2  47815  itcoval3  47816  itcovalsuc  47818
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