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Theorem tfr1 6658
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 2436 . . . 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 6644 . . 3  |-  Fun recs ( G )
31tfrlem14 6652 . . 3  |-  dom recs ( G )  =  On
4 df-fn 5457 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
52, 3, 4mpbir2an 887 . 2  |- recs ( G )  Fn  On
6 tfr.1 . . 3  |-  F  = recs ( G )
76fneq1i 5539 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
85, 7mpbir 201 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   {cab 2422   A.wral 2705   E.wrex 2706   Oncon0 4581   dom cdm 4878    |` cres 4880   Fun wfun 5448    Fn wfn 5449   ` cfv 5454  recscrecs 6632
This theorem is referenced by:  tfr2  6659  tfr3  6660  recsfnon  6661  rdgfnon  6676  dfac8alem  7910  dfac12lem1  8023  dfac12lem2  8024  zorn2lem1  8376  zorn2lem2  8377  zorn2lem4  8379  zorn2lem5  8380  zorn2lem6  8381  zorn2lem7  8382  ttukeylem3  8391  ttukeylem5  8393  ttukeylem6  8394  dnnumch1  27119  dnnumch3lem  27121  dnnumch3  27122  aomclem6  27134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633
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