Theorem itcoval1 45077

Description: A function iterated once. (Contributed by AV, 2-May-2024.)

Assertion

Ref	Expression
itcoval1	⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Proof of Theorem itcoval1

Dummy variables 𝑔 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itcoval 45075	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
2	1	fveq1d 6647	. . 3 ⊢ (𝐹 ∈ 𝑉 → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
3	2	adantl 485	. 2 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1))
4		nn0uz 12268	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
5		0nn0 11900	. . . . 5 ⊢ 0 ∈ ℕ0
6	5	a1i 11	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → 0 ∈ ℕ0)
7		1e0p1 12128	. . . 4 ⊢ 1 = (0 + 1)
8	1	eqcomd 2804	. . . . . . 7 ⊢ (𝐹 ∈ 𝑉 → seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))) = (IterComp‘𝐹))
9	8	fveq1d 6647	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ((IterComp‘𝐹)‘0))
10		itcoval0 45076	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
11	9, 10	eqtrd 2833	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
12	11	adantl 485	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘0) = ( I ↾ dom 𝐹))
13		eqidd 2799	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))
14		ax-1ne0 10595	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2989	. . . . . . . 8 ⊢ ¬ 1 = 0
16		eqeq1 2802	. . . . . . . 8 ⊢ (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0))
17	15, 16	mtbiri 330	. . . . . . 7 ⊢ (𝑖 = 1 → ¬ 𝑖 = 0)
18	17	iffalsed 4436	. . . . . 6 ⊢ (𝑖 = 1 → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
19	18	adantl 485	. . . . 5 ⊢ (((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑖 = 1) → if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹) = 𝐹)
20		1nn0 11901	. . . . . 6 ⊢ 1 ∈ ℕ0
21	20	a1i 11	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → 1 ∈ ℕ0)
22		simpr 488	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉)
23	13, 19, 21, 22	fvmptd 6752	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))‘1) = 𝐹)
24	4, 6, 7, 12, 23	seqp1d 13381	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹))
25		eqidd 2799	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)))
26		coeq2 5693	. . . . . . 7 ⊢ (𝑔 = ( I ↾ dom 𝐹) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
27	26	ad2antrl 727	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑔 = ( I ↾ dom 𝐹) ∧ 𝑗 = 𝐹)) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ( I ↾ dom 𝐹)))
28		dmexg 7594	. . . . . . 7 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
29	28	resiexd 6956	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ( I ↾ dom 𝐹) ∈ V)
30		elex 3459	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
31		coexg 7616	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ ( I ↾ dom 𝐹) ∈ V) → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
32	29, 31	mpdan 686	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∘ ( I ↾ dom 𝐹)) ∈ V)
33	25, 27, 29, 30, 32	ovmpod 7281	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
34	33	adantl 485	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹) = (𝐹 ∘ ( I ↾ dom 𝐹)))
35		coires1 6084	. . . . 5 ⊢ (𝐹 ∘ ( I ↾ dom 𝐹)) = (𝐹 ↾ dom 𝐹)
36		resdm 5863	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
37	36	adantr 484	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (𝐹 ↾ dom 𝐹) = 𝐹)
38	35, 37	syl5eq 2845	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (𝐹 ∘ ( I ↾ dom 𝐹)) = 𝐹)
39	34, 38	eqtrd 2833	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (( I ↾ dom 𝐹)(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹) = 𝐹)
40	24, 39	eqtrd 2833	. 2 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → (seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹)))‘1) = 𝐹)
41	3, 40	eqtrd 2833	1 ⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ifcif 4425   ↦ cmpt 5110   I cid 5424  dom cdm 5519   ↾ cres 5521   ∘ ccom 5523  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  0cc0 10526  1c1 10527  ℕ₀cn0 11885  seqcseq 13364  IterCompcitco 45071

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-itco 45073

This theorem is referenced by:  itcoval2  45078  ackvalsuc0val  45101