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Theorem tfrfun 6267
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 2157 . 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 6264 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   {cab 2143   A.wral 2435   E.wrex 2436   Oncon0 4323    |` cres 4588   Fun wfun 5164    Fn wfn 5165   ` cfv 5170  recscrecs 6251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178  df-recs 6252
This theorem is referenced by:  tfr1onlembfn  6291  tfr1onlemubacc  6293  tfri1dALT  6298  tfrcllembfn  6304  tfrcllemubacc  6306  tfrcl  6311  frecex  6341  frecfun  6342  frecfcllem  6351  frecsuclem  6353
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