Theorem orbitcl 46054

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 8448	. . . . 5 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
2		fvelrnb 6948	. . . . 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 8450	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
5		peano2 7903	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
6		fnfvelrn 7083	. . . . . . . 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 599	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
8	4, 7	eqeltrrd 2861	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
9		fveq2 6888	. . . . . . 7 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) = (𝐹‘𝐵))
10	9	eleq1d 2845	. . . . . 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 248	. . . . 5 ⊢ (𝑥 ∈ ω → (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω)))
12	11	rexlimiv 3156	. . . 4 ⊢ (∃𝑥 ∈ ω ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
13	3, 12	sylbi 220	. . 3 ⊢ (𝐵 ∈ ran (rec(𝐹, 𝐴) ↾ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
14		df-ima 5663	. . 3 ⊢ (rec(𝐹, 𝐴) “ ω) = ran (rec(𝐹, 𝐴) ↾ ω)
15	13, 14	eleq2s 2878	. 2 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
16	15, 14	eleqtrrdi 2871	1 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ran crn 5651   ↾ cres 5652   “ cima 5653  suc csuc 6366   Fn wfn 6536  ‘cfv 6541  ωcom 7879  reccrdg 8422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-om 7880  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423

This theorem is used by:  orbitclmpt  46055