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Theorem tfri1 6014
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 1379 . . . 4  |-  A. x
( Fun  G  /\  ( G `  x )  e.  _V )
43a1i 9 . . 3  |-  ( T. 
->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
51, 4tfri1d 5984 . 2  |-  ( T. 
->  F  Fn  On )
65trud 1294 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 102   A.wal 1283    = wceq 1285   T. wtru 1286    e. wcel 1434   _Vcvv 2602   Oncon0 4126   Fun wfun 4926    Fn wfn 4927   ` cfv 4932  recscrecs 5953
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-recs 5954
This theorem is referenced by:  tfri2  6015  tfri3  6016
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