Theorem orbitcl 45666

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 6941	. . . . 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 7075	. . . . . . . 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 2864	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
9		fveq2 6881	. . . . . . 7 ⊢ (((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝑥)) = (𝐹‘𝐵))
10	9	eleq1d 2848	. . . . . 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 3159	. . . 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 5674	. . 3 ⊢ (rec(𝐹, 𝐴) “ ω) = ran (rec(𝐹, 𝐴) ↾ ω)
15	13, 14	eleq2s 2881	. 2 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ ran (rec(𝐹, 𝐴) ↾ ω))
16	15, 14	eleqtrrdi 2874	1 ⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ran crn 5662   ↾ cres 5663   “ cima 5664  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7858  reccrdg 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393

This theorem is referenced by:  orbitclmpt  45667