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Theorem tfr3ALT 25304
Description: tfr3 6597 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
tfr3ALT  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Distinct variable groups:    f, F, x, y    f, G, x, y    x, B
Allowed substitution hints:    A( x, y, f)    B( y, f)

Proof of Theorem tfr3ALT
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5669 . . . . 5  |-  ( w  =  x  ->  ( B `  w )  =  ( B `  x ) )
2 predeq3 25196 . . . . . . 7  |-  ( w  =  x  ->  Pred (  _E  ,  On ,  w
)  =  Pred (  _E  ,  On ,  x
) )
32reseq2d 5087 . . . . . 6  |-  ( w  =  x  ->  ( B  |`  Pred (  _E  ,  On ,  w )
)  =  ( B  |`  Pred (  _E  ,  On ,  x )
) )
43fveq2d 5673 . . . . 5  |-  ( w  =  x  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  w )
) )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x )
) ) )
51, 4eqeq12d 2402 . . . 4  |-  ( w  =  x  ->  (
( B `  w
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  w ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) ) ) )
65cbvralv 2876 . . 3  |-  ( A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x )
) ) )
7 predon 25218 . . . . . . 7  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
87reseq2d 5087 . . . . . 6  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
98fveq2d 5673 . . . . 5  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
109eqeq2d 2399 . . . 4  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
1110ralbiia 2682 . . 3  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
126, 11bitri 241 . 2  |-  ( A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
13 epweon 4705 . . . 4  |-  _E  We  On
14 epse 4507 . . . 4  |-  _E Se  On
15 tfrALT.1 . . . . 5  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
16 tfrALTlem 25301 . . . . 5  |-  { 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 ) ) ) ) }
1715, 16eqtri 2408 . . . 4  |-  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 ) ) ) ) }
18 tfrALT.2 . . . 4  |-  F  = 
U. A
1913, 14, 17, 18wfr3 25300 . . 3  |-  ( ( B  Fn  On  /\  A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) ) )  ->  F  =  B )
2019eqcomd 2393 . 2  |-  ( ( B  Fn  On  /\  A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) ) )  ->  B  =  F )
2112, 20sylan2br 463 1  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   E.wrex 2651    C_ wss 3264   U.cuni 3958    _E cep 4434   Oncon0 4523    |` cres 4821    Fn wfn 5390   ` cfv 5395   Predcpred 25192
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-pred 25193
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