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Theorem itcoval 47604
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 47602 . 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 2727 . . 3 (𝑓 = 𝐹 β†’ 0 = 0)
3 coeq1 5850 . . . 4 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
43mpoeq3dv 7483 . . 3 (𝑓 = 𝐹 β†’ (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
5 dmeq 5896 . . . . . 6 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
65reseq2d 5974 . . . . 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 4544 . . . 4 (𝑓 = 𝐹 β†’ if(𝑖 = 0, ( I β†Ύ dom 𝑓), 𝑓) = if(𝑖 = 0, ( I β†Ύ dom 𝐹), 𝐹))
98mpteq2dv 5243 . . 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 13978 . 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 3487 . 2 (𝐹 ∈ 𝑉 β†’ 𝐹 ∈ V)
12 seqex 13971 . . 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 7014 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 3468  ifcif 4523   ↦ cmpt 5224   I cid 5566  dom cdm 5669   β†Ύ cres 5671   ∘ ccom 5673  β€˜cfv 6536   ∈ cmpo 7406  0cc0 11109  β„•0cn0 12473  seqcseq 13969  IterCompcitco 47600
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-seq 13970  df-itco 47602
This theorem is referenced by:  itcoval0  47605  itcoval1  47606  itcoval2  47607  itcoval3  47608  itcovalsuc  47610
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