Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tfr3ALT Unicode version

Theorem tfr3ALT 23690
Description: tfr3 6415 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 5525 . . . . 5  |-  ( w  =  x  ->  ( B `  w )  =  ( B `  x ) )
2 predeq3 23582 . . . . . . 7  |-  ( w  =  x  ->  Pred (  _E  ,  On ,  w
)  =  Pred (  _E  ,  On ,  x
) )
32reseq2d 4955 . . . . . 6  |-  ( w  =  x  ->  ( B  |`  Pred (  _E  ,  On ,  w )
)  =  ( B  |`  Pred (  _E  ,  On ,  x )
) )
43fveq2d 5529 . . . . 5  |-  ( w  =  x  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  w )
) )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x )
) ) )
51, 4eqeq12d 2297 . . . 4  |-  ( w  =  x  ->  (
( B `  w
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  w ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) ) ) )
65cbvralv 2764 . . 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 23604 . . . . . . 7  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
87reseq2d 4955 . . . . . 6  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
98fveq2d 5529 . . . . 5  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
109eqeq2d 2294 . . . 4  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
1110ralbiia 2575 . . 3  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
126, 11bitri 240 . 2  |-  ( A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) )  <->  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
13 epweon 4575 . . . 4  |-  _E  We  On
14 epse 4376 . . . 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 23687 . . . . 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 2303 . . . 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 23686 . . 3  |-  ( ( B  Fn  On  /\  A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) ) )  ->  F  =  B )
2019eqcomd 2288 . 2  |-  ( ( B  Fn  On  /\  A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) ) )  ->  B  =  F )
2112, 20sylan2br 462 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 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827    _E cep 4303   Oncon0 4392    |` cres 4691    Fn wfn 5250   ` cfv 5255   Predcpred 23578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-pred 23579
  Copyright terms: Public domain W3C validator