Theorem orbitcl 45401

Description: The orbit under a function is closed under the function. (Contributed by Eric Schmidt, 6-Nov-2025.)

Assertion

Ref	Expression
orbitcl	⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

Proof of Theorem orbitcl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8364	. . . . 5 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
2		fvelrnb 6887	. . . . 5 ⊢ ((rec(𝐹, 𝐴) ↾ ω) Fn ω → (𝐵 ∈ ran (rec(𝐹, 𝐴) ↾ ω) ↔ ∃𝑥 ∈ ω ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵))
3	1, 2	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ ran (rec(𝐹, 𝐴) ↾ ω) ↔ ∃𝑥 ∈ ω ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵)
4		frsuc 8366	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
5		peano2 7830	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
6		fnfvelrn 7021	. . . . . . . 8 ⊢ (((rec(𝐹, 𝐴) ↾ ω) Fn ω ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
7	1, 5, 6	sylancr 593	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
8	4, 7	eqeltrrd 2840	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
9		fveq2 6827	. . . . . . 7 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) = (𝐹‘𝐵))
10	9	eleq1d 2824	. . . . . 6 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → ((𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω) ↔ (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω)))
11	8, 10	syl5ibcom 246	. . . . 5 ⊢ (𝑥 ∈ ω → (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω)))
12	11	rexlimiv 3133	. . . 4 ⊢ (∃𝑥 ∈ ω ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
13	3, 12	sylbi 218	. . 3 ⊢ (𝐵 ∈ ran (rec(𝐹, 𝐴) ↾ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
14		df-ima 5631	. . 3 ⊢ (rec(𝐹, 𝐴) “ ω) = ran (rec(𝐹, 𝐴) ↾ ω)
15	13, 14	eleq2s 2857	. 2 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
16	15, 14	eleqtrrdi 2850	1 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  ran crn 5619   ↾ cres 5620   “ cima 5621  suc csuc 6312   Fn wfn 6480  ‘cfv 6485  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  orbitclmpt  45402