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Theorem frecfnom 5919
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 5918 . . . 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 5887 . . 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 4954 . . 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 4277 . . . . . 6 𝜔 ⊆ On
8 sseqin2 3150 . . . . . 6 (𝜔 ⊆ On ↔ (On ∩ 𝜔) = 𝜔)
97, 8mpbi 133 . . . . 5 (On ∩ 𝜔) = 𝜔
109reseq2i 4551 . . . 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 5915 . . . 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 4933 . . 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 3808  Oncon0 4065  suc csuc 4067  𝜔com 4255  dom cdm 4287  cres 4289   Fn wfn 4839  cfv 4844  recscrecs 5857  freccfrec 5914
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-bnd 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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-recs 5858  df-frec 5915
This theorem is referenced by:  frecrdg  5925  frec2uzrand  8818  frec2uzf1od  8819  frecfzennn  8822
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