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Theorem frecfun 6244
 Description: Finite recursion produces a function. See also frecfnom 6250 which also states that the domain of that function is but which puts conditions on and . (Contributed by Jim Kingdon, 13-Feb-2022.)
Assertion
Ref Expression
frecfun frec

Proof of Theorem frecfun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrfun 6169 . . 3 recs
2 funres 5120 . . 3 recs recs
31, 2ax-mp 7 . 2 recs
4 df-frec 6240 . . 3 frec recs
54funeqi 5100 . 2 frec recs
63, 5mpbir 145 1 frec
 Colors of variables: wff set class Syntax hints:   wa 103   wo 680   wceq 1312   wcel 1461  cab 2099  wrex 2389  cvv 2655  c0 3327   cmpt 3947   csuc 4245  com 4462   cdm 4497   cres 4499   wfun 5073  cfv 5079  recscrecs 6153  freccfrec 6239 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-res 4509  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087  df-recs 6154  df-frec 6240 This theorem is referenced by: (None)
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