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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(𝐹z) 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.
StepHypRef Expression
1 eqid 2037 . . . 4 recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
2 eqid 2037 . . . . 5 (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
32frectfr 5924 . . . 4 ((z(𝐹z) V A 𝑉) → y(Fun (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘y) V))
41, 3tfri1d 5890 . . 3 ((z(𝐹z) V A 𝑉) → recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) Fn On)
5 fnresin1 4956 . . 3 (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) Fn On → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔))
64, 5syl 14 . 2 ((z(𝐹z) V A 𝑉) → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔))
7 omsson 4278 . . . . . 6 𝜔 ⊆ On
8 sseqin2 3150 . . . . . 6 (𝜔 ⊆ On ↔ (On ∩ 𝜔) = 𝜔)
97, 8mpbi 133 . . . . 5 (On ∩ 𝜔) = 𝜔
109reseq2i 4552 . . . 4 (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (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‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
1210, 11eqtr4i 2060 . . 3 (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (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‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) = frec(𝐹, A) (On ∩ 𝜔) = 𝜔) → ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔) ↔ frec(𝐹, A) Fn 𝜔))
1412, 9, 13mp2an 402 . 2 ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ (On ∩ 𝜔)) Fn (On ∩ 𝜔) ↔ frec(𝐹, A) Fn 𝜔)
156, 14sylib 127 1 ((z(𝐹z) V A 𝑉) → frec(𝐹, A) Fn 𝜔)
 Colors of variables: wff set class 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
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