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Theorem tfri1 6230
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 1410 . . . 4  |-  A. x
( Fun  G  /\  ( G `  x )  e.  _V )
43a1i 9 . . 3  |-  ( T. 
->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
51, 4tfri1d 6200 . 2  |-  ( T. 
->  F  Fn  On )
65mptru 1325 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 103   A.wal 1314    = wceq 1316   T. wtru 1317    e. wcel 1465   _Vcvv 2660   Oncon0 4255   Fun wfun 5087    Fn wfn 5088   ` cfv 5093  recscrecs 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfri2  6231  tfri3  6232
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