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Theorem tfri1 6122
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
tfri1.1  |-  F  = recs ( G )
tfri1.2  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
Assertion
Ref Expression
tfri1  |-  F  Fn  On
Distinct variable group:    x, G
Allowed substitution hint:    F( x)

Proof of Theorem tfri1
StepHypRef Expression
1 tfri1.1 . . 3  |-  F  = recs ( G )
2 tfri1.2 . . . . 5  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
32ax-gen 1383 . . . 4  |-  A. x
( Fun  G  /\  ( G `  x )  e.  _V )
43a1i 9 . . 3  |-  ( T. 
->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
51, 4tfri1d 6092 . 2  |-  ( T. 
->  F  Fn  On )
65mptru 1298 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 102   A.wal 1287    = wceq 1289   T. wtru 1290    e. wcel 1438   _Vcvv 2619   Oncon0 4188   Fun wfun 5004    Fn wfn 5005   ` cfv 5010  recscrecs 6061
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 3952  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351
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 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-recs 6062
This theorem is referenced by:  tfri2  6123  tfri3  6124
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