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Theorem wfr1 7059
Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G. 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, 3wfrfun 7051 . 2  |-  Fun  F
5 eqid 2422 . . 3  |-  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
61, 2, 3, 5wfrlem16 7056 . 2  |-  dom  F  =  A
7 df-fn 5601 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
84, 6, 7mpbir2an 928 1  |-  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3434   {csn 3996   <.cop 4002   Se wse 4807    We wwe 4808   dom cdm 4850    |` cres 4852   Predcpred 5395   Fun wfun 5592    Fn wfn 5593   ` cfv 5598  wrecscwrecs 7032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-wrecs 7033
This theorem is referenced by:  wfr3  7061  tfr1ALT  7123  bpolylem  14089
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