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Theorem wfr1 27879
Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G and a class of "acceptable" functions  B. Then, using a base class  A and a well-ordering  R of  A, we define a function  F. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of  F. We begin by showing that  F is a function over  A. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr1.1  |-  R  We  A
wfr1.2  |-  R Se  A
wfr1.3  |-  F  = wrecs ( R ,  A ,  G )
Assertion
Ref Expression
wfr1  |-  F  Fn  A

Proof of Theorem wfr1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 wfr1.1 . . 3  |-  R  We  A
2 wfr1.2 . . 3  |-  R Se  A
3 wfr1.3 . . 3  |-  F  = wrecs ( R ,  A ,  G )
41, 2, 3wfrlem11 27873 . 2  |-  Fun  F
5 eqid 2452 . . 3  |-  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
61, 2, 3, 5wfrlem16 27878 . 2  |-  dom  F  =  A
7 df-fn 5524 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
84, 6, 7mpbir2an 911 1  |-  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    u. cun 3429   {csn 3980   <.cop 3986   Se wse 4780    We wwe 4781   dom cdm 4943    |` cres 4945   Fun wfun 5515    Fn wfn 5516   ` cfv 5521   Predcpred 27763  wrecscwrecs 27855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-pred 27764  df-wrecs 27856
This theorem is referenced by:  wfr3  27881  tfrALTlem  27882  tfr1ALT  27883  bpolylem  28330
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