Theorem frecfnom 5925

Description: The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)

Assertion

Ref	Expression
frecfnom	⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → frec(𝐹, A) Fn 𝜔)

Distinct variable groups:   z,A   z,𝐹

Allowed substitution hint:   𝑉(z)

Proof of Theorem frecfnom

Dummy variables g 𝑚 x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2037	. . . 4 ⊢ recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) = recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))}))
2		eqid 2037	. . . . 5 ⊢ (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))}) = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})
3	2	frectfr 5924	. . . 4 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀y(Fun (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))}) ∧ ((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})‘y) ∈ V))
4	1, 3	tfri1d 5890	. . 3 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) Fn On)
5		fnresin1 4956	. . 3 ⊢ (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) Fn On → (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔))
6	4, 5	syl 14	. 2 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔))
7		omsson 4278	. . . . . 6 ⊢ 𝜔 ⊆ On
8		sseqin2 3150	. . . . . 6 ⊢ (𝜔 ⊆ On ↔ (On ∩ 𝜔) = 𝜔)
9	7, 8	mpbi 133	. . . . 5 ⊢ (On ∩ 𝜔) = 𝜔
10	9	reseq2i 4552	. . . 4 ⊢ (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) = (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ 𝜔)
11		df-frec 5918	. . . 4 ⊢ frec(𝐹, A) = (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ 𝜔)
12	10, 11	eqtr4i 2060	. . 3 ⊢ (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) = frec(𝐹, A)
13		fneq12 4935	. . 3 ⊢ (((recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) = frec(𝐹, A) ∧ (On ∩ 𝜔) = 𝜔) → ((recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔) ↔ frec(𝐹, A) Fn 𝜔))
14	12, 9, 13	mp2an 402	. 2 ⊢ ((recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔) ↔ frec(𝐹, A) Fn 𝜔)
15	6, 14	sylib 127	1 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → frec(𝐹, A) Fn 𝜔)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218   ↦ cmpt 3809  Oncon0 4066  suc csuc 4068  𝜔com 4256  dom cdm 4288   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845  recscrecs 5860  freccfrec 5917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-frec 5918

This theorem is referenced by:  frecrdg  5931  freccl  5932  frec2uzrand  8872  frec2uzf1od  8873  frecuzrdgrom  8877  frecfzennn  8884