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Theorem frfnom 8064
Description: The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
frfnom (rec(𝐹, 𝐴) ↾ ω) Fn ω

Proof of Theorem frfnom
StepHypRef Expression
1 rdgfun 8046 . . 3 Fun rec(𝐹, 𝐴)
2 funres 6393 . . 3 (Fun rec(𝐹, 𝐴) → Fun (rec(𝐹, 𝐴) ↾ ω))
31, 2ax-mp 5 . 2 Fun (rec(𝐹, 𝐴) ↾ ω)
4 dmres 5873 . . 3 dom (rec(𝐹, 𝐴) ↾ ω) = (ω ∩ dom rec(𝐹, 𝐴))
5 rdgdmlim 8047 . . . . 5 Lim dom rec(𝐹, 𝐴)
6 limomss 7576 . . . . 5 (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴))
75, 6ax-mp 5 . . . 4 ω ⊆ dom rec(𝐹, 𝐴)
8 df-ss 3955 . . . 4 (ω ⊆ dom rec(𝐹, 𝐴) ↔ (ω ∩ dom rec(𝐹, 𝐴)) = ω)
97, 8mpbi 231 . . 3 (ω ∩ dom rec(𝐹, 𝐴)) = ω
104, 9eqtri 2848 . 2 dom (rec(𝐹, 𝐴) ↾ ω) = ω
11 df-fn 6354 . 2 ((rec(𝐹, 𝐴) ↾ ω) Fn ω ↔ (Fun (rec(𝐹, 𝐴) ↾ ω) ∧ dom (rec(𝐹, 𝐴) ↾ ω) = ω))
123, 10, 11mpbir2an 707 1 (rec(𝐹, 𝐴) ↾ ω) Fn ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cin 3938  wss 3939  dom cdm 5553  cres 5555  Lim wlim 6189  Fun wfun 6345   Fn wfn 6346  ωcom 7571  reccrdg 8039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040
This theorem is referenced by:  frsucmptn  8068  seqomlem2  8081  seqomlem3  8082  seqomlem4  8083  unblem4  8765  dffi3  8887  inf0  9076  inf3lem6  9088  alephfplem4  9525  alephfp  9526  infpssrlem3  9719  itunifn  9831  hsmexlem5  9844  axdclem2  9934  wunex2  10152  wuncval2  10161  peano5nni  11633  1nn  11641  peano2nn  11642  om2uzrani  13313  om2uzf1oi  13314  uzrdglem  13318  uzrdgfni  13319  uzrdg0i  13320  hashkf  13685  hashgval2  13732  dftrpred2  32943  trpredpred  32952  trpredex  32961  neibastop2lem  33593
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