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Theorem tfr1 6299
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
StepHypRef Expression
1 eqid 2253 . . . 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 6285 . . 3  |-  Fun recs ( G )
31tfrlem14 6293 . . 3  |-  dom recs ( G )  =  On
4 df-fn 4603 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
52, 3, 4mpbir2an 891 . 2  |- recs ( G )  Fn  On
6 tfr.1 . . 3  |-  F  = recs ( G )
76fneq1i 5195 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
85, 7mpbir 202 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1619   {cab 2239   A.wral 2509   E.wrex 2510   Oncon0 4285   dom cdm 4580    |` cres 4582   Fun wfun 4586    Fn wfn 4587   ` cfv 4592  recscrecs 6273
This theorem is referenced by:  tfr2  6300  tfr3  6301  recsfnon  6302  rdgfnon  6317  dfac8alem  7540  dfac12lem1  7653  dfac12lem2  7654  zorn2lem1  8007  zorn2lem2  8008  zorn2lem4  8010  zorn2lem5  8011  zorn2lem6  8012  zorn2lem7  8013  ttukeylem3  8022  ttukeylem5  8024  ttukeylem6  8025  dnnumch1  26307  dnnumch3lem  26309  dnnumch3  26310  aomclem6  26322
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274
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