Theorem orbitcl 45553

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 8418	. . . . 5 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
2		fvelrnb 6939	. . . . 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 8420	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
5		peano2 7882	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
6		fnfvelrn 7073	. . . . . . . 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 598	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑥) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
8	4, 7	eqeltrrd 2870	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
9		fveq2 6879	. . . . . . 7 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) = (𝐹‘𝐵))
10	9	eleq1d 2854	. . . . . 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 3165	. . . 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 5672	. . 3 ⊢ (rec(𝐹, 𝐴) “ ω) = ran (rec(𝐹, 𝐴) ↾ ω)
15	13, 14	eleq2s 2887	. 2 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
16	15, 14	eleqtrrdi 2880	1 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  ran crn 5660   ↾ cres 5661   “ cima 5662  suc csuc 6360   Fn wfn 6529  ‘cfv 6534  ωcom 7858  reccrdg 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393

This theorem is referenced by:  orbitclmpt  45554