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Theorem itcoval 47300
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 47298 . 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 2733 . . 3 (𝑓 = 𝐹 β†’ 0 = 0)
3 coeq1 5855 . . . 4 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
43mpoeq3dv 7484 . . 3 (𝑓 = 𝐹 β†’ (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5 dmeq 5901 . . . . . 6 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
65reseq2d 5979 . . . . 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 4548 . . . 4 (𝑓 = 𝐹 β†’ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))
98mpteq2dv 5249 . . 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 13971 . 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 13964 . . 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 7018 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 1541   ∈ wcel 2106  Vcvv 3474  ifcif 4527   ↦ cmpt 5230   I cid 5572  dom cdm 5675   β†Ύ cres 5677   ∘ ccom 5679  β€˜cfv 6540   ∈ cmpo 7407  0cc0 11106  β„•0cn0 12468  seqcseq 13962  IterCompcitco 47296
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seq 13963  df-itco 47298
This theorem is referenced by:  itcoval0  47301  itcoval1  47302  itcoval2  47303  itcoval3  47304  itcovalsuc  47306
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