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Theorem tfr3ALT 27715
Description: tfr3 6850 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 27623 . . . . . 6  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
21reseq2d 5105 . . . . 5  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
32fveq2d 5690 . . . 4  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
43eqeq2d 2449 . . 3  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
54ralbiia 2742 . 2  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
6 epweon 6390 . . . 4  |-  _E  We  On
7 epse 4698 . . . 4  |-  _E Se  On
8 tfrALT.1 . . . . 5  |-  F  = recs ( G )
9 tfrALTlem 27712 . . . . 5  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
108, 9eqtri 2458 . . . 4  |-  F  = wrecs (  _E  ,  On ,  G )
116, 7, 10wfr3 27711 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) ) )  ->  F  =  B )
1211eqcomd 2443 . 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 1369    e. wcel 1756   A.wral 2710    _E cep 4625   Oncon0 4714    |` cres 4837    Fn wfn 5408   ` cfv 5413  recscrecs 6823   Predcpred 27593  wrecscwrecs 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6824  df-pred 27594  df-wrecs 27686
This theorem is referenced by: (None)
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