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Theorem tfrfun 6067
Description: Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
Assertion
Ref Expression
tfrfun  |-  Fun recs ( F )

Proof of Theorem tfrfun
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2088 . 2  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
21tfrlem7 6064 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   {cab 2074   A.wral 2359   E.wrex 2360   Oncon0 4181    |` cres 4430   Fun wfun 4996    Fn wfn 4997   ` cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-recs 6052
This theorem is referenced by:  tfr1onlembfn  6091  tfr1onlemubacc  6093  tfri1dALT  6098  tfrcllembfn  6104  tfrcllemubacc  6106  tfrcl  6111  frecex  6141  frecfun  6142  frecfcllem  6151  frecsuclem  6153
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