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Theorem wfr1 29602
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 29596 . 2  |-  Fun  F
5 eqid 2454 . . 3  |-  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
61, 2, 3, 5wfrlem16 29601 . 2  |-  dom  F  =  A
7 df-fn 5573 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
84, 6, 7mpbir2an 918 1  |-  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    u. cun 3459   {csn 4016   <.cop 4022   Se wse 4825    We wwe 4826   dom cdm 4988    |` cres 4990   Fun wfun 5564    Fn wfn 5565   ` cfv 5570   Predcpred 29486  wrecscwrecs 29578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-pred 29487  df-wrecs 29579
This theorem is referenced by:  wfr3  29604  tfrALTlem  29605  tfr1ALT  29606  bpolylem  30041
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