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Theorem tfr1 7058
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class  G, normally a function, and define a class  A of all "acceptable" functions. The final function we're interested in is the union  F  = recs ( G ) of them.  F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of  F. In this first part we show that  F is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr1  |-  F  Fn  On

Proof of Theorem tfr1
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . 4  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
21tfrlem7 7044 . . 3  |-  Fun recs ( G )
31tfrlem14 7052 . . 3  |-  dom recs ( G )  =  On
4 df-fn 5573 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
52, 3, 4mpbir2an 918 . 2  |- recs ( G )  Fn  On
6 tfr.1 . . 3  |-  F  = recs ( G )
76fneq1i 5657 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
85, 7mpbir 209 1  |-  F  Fn  On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   {cab 2439   A.wral 2804   E.wrex 2805   Oncon0 4867   dom cdm 4988    |` cres 4990   Fun wfun 5564    Fn wfn 5565   ` cfv 5570  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-recs 7034
This theorem is referenced by:  tfr2  7059  tfr3  7060  recsfnon  7061  rdgfnon  7076  dfac8alem  8401  dfac12lem1  8514  dfac12lem2  8515  zorn2lem1  8867  zorn2lem2  8868  zorn2lem4  8870  zorn2lem5  8871  zorn2lem6  8872  zorn2lem7  8873  ttukeylem3  8882  ttukeylem5  8884  ttukeylem6  8885  dnnumch1  31232  dnnumch3lem  31234  dnnumch3  31235  aomclem6  31247
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