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Theorem tfrfun 6020
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 2085 . 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 6017 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1287   {cab 2071   A.wral 2355   E.wrex 2356   Oncon0 4157    |` cres 4406   Fun wfun 4966    Fn wfn 4967   ` cfv 4972  recscrecs 6004
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-res 4416  df-iota 4937  df-fun 4974  df-fn 4975  df-fv 4980  df-recs 6005
This theorem is referenced by:  tfr1onlembfn  6044  tfr1onlemubacc  6046  tfri1dALT  6051  tfrcllembfn  6057  tfrcllemubacc  6059  tfrcl  6064  frecex  6094  frecfun  6095  frecfcllem  6104  frecsuclem  6106
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