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Theorem tfr2ALT 25492
Description: tfr2 6618 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
tfrALT.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfrALT.2  |-  F  = 
U. A
Assertion
Ref Expression
tfr2ALT  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Distinct variable groups:    f, F, x, y    f, G, x, y    y, z
Allowed substitution hints:    A( x, y, z, f)    F( z)    G( z)

Proof of Theorem tfr2ALT
StepHypRef Expression
1 epweon 4723 . . 3  |-  _E  We  On
2 epse 4525 . . 3  |-  _E Se  On
3 tfrALT.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
4 tfrALTlem 25490 . . . 4  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  =  { f  |  E. x ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) }
53, 4eqtri 2424 . . 3  |-  A  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) }
6 tfrALT.2 . . 3  |-  F  = 
U. A
71, 2, 5, 6wfr2 25487 . 2  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  Pred (  _E  ,  On ,  z ) ) ) )
8 predon 25407 . . . 4  |-  ( z  e.  On  ->  Pred (  _E  ,  On ,  z )  =  z )
98reseq2d 5105 . . 3  |-  ( z  e.  On  ->  ( F  |`  Pred (  _E  ,  On ,  z )
)  =  ( F  |`  z ) )
109fveq2d 5691 . 2  |-  ( z  e.  On  ->  ( G `  ( F  |` 
Pred (  _E  ,  On ,  z )
) )  =  ( G `  ( F  |`  z ) ) )
117, 10eqtrd 2436 1  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    C_ wss 3280   U.cuni 3975    _E cep 4452   Oncon0 4541    |` cres 4839    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-pred 25382
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