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Theorem tfrfun 6320
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 2177 . 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 6317 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   {cab 2163   A.wral 2455   E.wrex 2456   Oncon0 4363    |` cres 4628   Fun wfun 5210    Fn wfn 5211   ` cfv 5216  recscrecs 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-recs 6305
This theorem is referenced by:  tfr1onlembfn  6344  tfr1onlemubacc  6346  tfri1dALT  6351  tfrcllembfn  6357  tfrcllemubacc  6359  tfrcl  6364  frecex  6394  frecfun  6395  frecfcllem  6404  frecsuclem  6406
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