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Theorem wfr2 25547
Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of  F at any  z  e.  A is  G recursively applied to all "previous" values of  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr2.1  |-  R  We  A
wfr2.2  |-  R Se  A
wfr2.3  |-  F  = wrecs ( R ,  A ,  G )
Assertion
Ref Expression
wfr2  |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )

Proof of Theorem wfr2
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . 3  |-  ( y  =  X  ->  ( F `  y )  =  ( F `  X ) )
2 predeq3 25435 . . . . 5  |-  ( y  =  X  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  X ) )
32reseq2d 5138 . . . 4  |-  ( y  =  X  ->  ( F  |`  Pred ( R ,  A ,  y )
)  =  ( F  |`  Pred ( R ,  A ,  X )
) )
43fveq2d 5724 . . 3  |-  ( y  =  X  ->  ( G `  ( F  |` 
Pred ( R ,  A ,  y )
) )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X )
) ) )
51, 4eqeq12d 2449 . 2  |-  ( y  =  X  ->  (
( F `  y
)  =  ( G `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  <->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X )
) ) ) )
6 wfr2.1 . . . . 5  |-  R  We  A
7 wfr2.2 . . . . 5  |-  R Se  A
8 wfr2.3 . . . . 5  |-  F  = wrecs ( R ,  A ,  G )
9 eqid 2435 . . . . 5  |-  ( F  u.  { <. w ,  ( G `  ( F  |`  Pred ( R ,  A ,  w ) ) )
>. } )  =  ( F  u.  { <. w ,  ( G `  ( F  |`  Pred ( R ,  A ,  w ) ) )
>. } )
106, 7, 8, 9wfrlem16 25545 . . . 4  |-  dom  F  =  A
1110eleq2i 2499 . . 3  |-  ( y  e.  dom  F  <->  y  e.  A )
126, 7, 8wfrlem12 25541 . . 3  |-  ( y  e.  dom  F  -> 
( F `  y
)  =  ( G `
 ( F  |`  Pred ( R ,  A ,  y ) ) ) )
1311, 12sylbir 205 . 2  |-  ( y  e.  A  ->  ( F `  y )  =  ( G `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )
145, 13vtoclga 3009 1  |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    u. cun 3310   {csn 3806   <.cop 3809   Se wse 4531    We wwe 4532   dom cdm 4870    |` cres 4872   ` cfv 5446   Predcpred 25430  wrecscwrecs 25522
This theorem is referenced by:  wfr3  25548  tfrALTlem  25549  tfr2ALT  25551  bpolylem  26086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-pred 25431  df-wrecs 25523
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