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Theorem itcoval1 45030
 Description: A function iterated once. (Contributed by AV, 2-May-2024.)
Assertion
Ref Expression
itcoval1 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Proof of Theorem itcoval1
Dummy variables 𝑔 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itcoval 45028 . . . 4 (𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
21fveq1d 6665 . . 3 (𝐹𝑉 → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
32adantl 485 . 2 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
4 nn0uz 12279 . . . 4 0 = (ℤ‘0)
5 0nn0 11911 . . . . 5 0 ∈ ℕ0
65a1i 11 . . . 4 ((Rel 𝐹𝐹𝑉) → 0 ∈ ℕ0)
7 1e0p1 12139 . . . 4 1 = (0 + 1)
81eqcomd 2830 . . . . . . 7 (𝐹𝑉 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) = (IterComp‘𝐹))
98fveq1d 6665 . . . . . 6 (𝐹𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ((IterComp‘𝐹)‘0))
10 itcoval0 45029 . . . . . 6 (𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
119, 10eqtrd 2859 . . . . 5 (𝐹𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
1211adantl 485 . . . 4 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
13 eqidd 2825 . . . . 5 ((Rel 𝐹𝐹𝑉) → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))
14 ax-1ne0 10606 . . . . . . . . 9 1 ≠ 0
1514neii 3016 . . . . . . . 8 ¬ 1 = 0
16 eqeq1 2828 . . . . . . . 8 (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0))
1715, 16mtbiri 330 . . . . . . 7 (𝑖 = 1 → ¬ 𝑖 = 0)
1817iffalsed 4461 . . . . . 6 (𝑖 = 1 → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
1918adantl 485 . . . . 5 (((Rel 𝐹𝐹𝑉) ∧ 𝑖 = 1) → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
20 1nn0 11912 . . . . . 6 1 ∈ ℕ0
2120a1i 11 . . . . 5 ((Rel 𝐹𝐹𝑉) → 1 ∈ ℕ0)
22 simpr 488 . . . . 5 ((Rel 𝐹𝐹𝑉) → 𝐹𝑉)
2313, 19, 21, 22fvmptd 6768 . . . 4 ((Rel 𝐹𝐹𝑉) → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))‘1) = 𝐹)
244, 6, 7, 12, 23seqp1d 13392 . . 3 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹))
25 eqidd 2825 . . . . . 6 (𝐹𝑉 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)))
26 coeq2 5717 . . . . . . 7 (𝑔 = ( I ↾ dom 𝐹) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
2726ad2antrl 727 . . . . . 6 ((𝐹𝑉 ∧ (𝑔 = ( I ↾ dom 𝐹) ∧ 𝑗 = 𝐹)) → (𝐹𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
28 dmexg 7610 . . . . . . 7 (𝐹𝑉 → dom 𝐹 ∈ V)
2928resiexd 6972 . . . . . 6 (𝐹𝑉 → ( I ↾ dom 𝐹) ∈ V)
30 elex 3498 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 coexg 7631 . . . . . . 7 ((𝐹𝑉 ∧ ( I ↾ dom 𝐹) ∈ V) → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
3229, 31mpdan 686 . . . . . 6 (𝐹𝑉 → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
3325, 27, 29, 30, 32ovmpod 7297 . . . . 5 (𝐹𝑉 → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
3433adantl 485 . . . 4 ((Rel 𝐹𝐹𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
35 coires1 6106 . . . . 5 (𝐹 ∘ ( I ↾ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
36 resdm 5886 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
3736adantr 484 . . . . 5 ((Rel 𝐹𝐹𝑉) → (𝐹 ↾ dom 𝐹) = 𝐹)
3835, 37syl5eq 2871 . . . 4 ((Rel 𝐹𝐹𝑉) → (𝐹 ∘ ( I ↾ dom 𝐹)) = 𝐹)
3934, 38eqtrd 2859 . . 3 ((Rel 𝐹𝐹𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹) = 𝐹)
4024, 39eqtrd 2859 . 2 ((Rel 𝐹𝐹𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = 𝐹)
413, 40eqtrd 2859 1 ((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ifcif 4450   ↦ cmpt 5133   I cid 5447  dom cdm 5543   ↾ cres 5545   ∘ ccom 5547  Rel wrel 5548  ‘cfv 6345  (class class class)co 7151   ∈ cmpo 7153  0cc0 10537  1c1 10538  ℕ0cn0 11896  seqcseq 13375  IterCompcitco 45024 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 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614 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-nel 3119  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 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-n0 11897  df-z 11981  df-uz 12243  df-seq 13376  df-itco 45026 This theorem is referenced by:  itcoval2  45031  ackvalsuc0val  45054
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