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Theorem tfri1d 6100
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that  G is defined "everywhere", which is stated here as  ( G `  x )  e.  _V. Alternately,  A. x  e.  On A. f ( f  Fn  x  -> 
f  e.  dom  G
) would suffice.

Given a function  G satisfying that condition, we 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 Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

Hypotheses
Ref Expression
tfri1d.1  |-  F  = recs ( G )
tfri1d.2  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
Assertion
Ref Expression
tfri1d  |-  ( ph  ->  F  Fn  On )
Distinct variable group:    x, G
Allowed substitution hints:    ph( x)    F( x)

Proof of Theorem tfri1d
Dummy variables  f  g  u  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2088 . . . . . 6  |-  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }
21tfrlem3 6076 . . . . 5  |-  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
3 tfri1d.2 . . . . 5  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
42, 3tfrlemi14d 6098 . . . 4  |-  ( ph  ->  dom recs ( G )  =  On )
5 eqid 2088 . . . . 5  |-  { w  |  E. y  e.  On  ( w  Fn  y  /\  A. z  e.  y  ( w `  z
)  =  ( G `
 ( w  |`  z ) ) ) }  =  { w  |  E. y  e.  On  ( w  Fn  y  /\  A. z  e.  y  ( w `  z
)  =  ( G `
 ( w  |`  z ) ) ) }
65tfrlem7 6082 . . . 4  |-  Fun recs ( G )
74, 6jctil 305 . . 3  |-  ( ph  ->  ( Fun recs ( G
)  /\  dom recs ( G )  =  On ) )
8 df-fn 5018 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
97, 8sylibr 132 . 2  |-  ( ph  -> recs ( G )  Fn  On )
10 tfri1d.1 . . 3  |-  F  = recs ( G )
1110fneq1i 5108 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
129, 11sylibr 132 1  |-  ( ph  ->  F  Fn  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619   Oncon0 4190   dom cdm 4438    |` cres 4440   Fun wfun 5009    Fn wfn 5010   ` cfv 5015  recscrecs 6069
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-recs 6070
This theorem is referenced by:  tfri2d  6101  tfri1  6130  rdgifnon  6144  rdgifnon2  6145  frecfnom  6166
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