Theorem orbitcl 44931

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 8357	. . . . 5 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
2		fvelrnb 6883	. . . . 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 8359	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
5		peano2 7823	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
6		fnfvelrn 7014	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
8	4, 7	eqeltrrd 2829	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
9		fveq2 6822	. . . . . . 7 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) = (𝐹‘𝐵))
10	9	eleq1d 2813	. . . . . 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 245	. . . . 5 ⊢ (𝑥 ∈ ω → (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω)))
12	11	rexlimiv 3123	. . . 4 ⊢ (∃𝑥 ∈ ω ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
13	3, 12	sylbi 217	. . 3 ⊢ (𝐵 ∈ ran (rec(𝐹, 𝐴) ↾ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
14		df-ima 5632	. . 3 ⊢ (rec(𝐹, 𝐴) “ ω) = ran (rec(𝐹, 𝐴) ↾ ω)
15	13, 14	eleq2s 2846	. 2 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
16	15, 14	eleqtrrdi 2839	1 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ran crn 5620   ↾ cres 5621   “ cima 5622  suc csuc 6309   Fn wfn 6477  ‘cfv 6482  ωcom 7799  reccrdg 8331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332

This theorem is referenced by:  orbitclmpt  44932