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Theorem tfr3ALT 27889
Description: tfr3 6967 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
tfrALT.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr3ALT  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Distinct variable groups:    x, F    x, G    x, B

Proof of Theorem tfr3ALT
StepHypRef Expression
1 predon 27797 . . . . . 6  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
21reseq2d 5217 . . . . 5  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
32fveq2d 5802 . . . 4  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
43eqeq2d 2468 . . 3  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
54ralbiia 2837 . 2  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
6 epweon 6504 . . . 4  |-  _E  We  On
7 epse 4810 . . . 4  |-  _E Se  On
8 tfrALT.1 . . . . 5  |-  F  = recs ( G )
9 tfrALTlem 27886 . . . . 5  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
108, 9eqtri 2483 . . . 4  |-  F  = wrecs (  _E  ,  On ,  G )
116, 7, 10wfr3 27885 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) ) )  ->  F  =  B )
1211eqcomd 2462 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) ) )  ->  B  =  F )
135, 12sylan2br 476 1  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2798    _E cep 4737   Oncon0 4826    |` cres 4949    Fn wfn 5520   ` cfv 5525  recscrecs 6940   Predcpred 27767  wrecscwrecs 27859
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-recs 6941  df-pred 27768  df-wrecs 27860
This theorem is referenced by: (None)
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