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Theorem tfr3ALT 24350
Description: tfr3 6431 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 5541 . . . . 5  |-  ( w  =  x  ->  ( B `  w )  =  ( B `  x ) )
2 predeq3 24242 . . . . . . 7  |-  ( w  =  x  ->  Pred (  _E  ,  On ,  w
)  =  Pred (  _E  ,  On ,  x
) )
32reseq2d 4971 . . . . . 6  |-  ( w  =  x  ->  ( B  |`  Pred (  _E  ,  On ,  w )
)  =  ( B  |`  Pred (  _E  ,  On ,  x )
) )
43fveq2d 5545 . . . . 5  |-  ( w  =  x  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  w )
) )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  x )
) ) )
51, 4eqeq12d 2310 . . . 4  |-  ( w  =  x  ->  (
( B `  w
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  w ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) ) ) )
65cbvralv 2777 . . 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 24264 . . . . . . 7  |-  ( x  e.  On  ->  Pred (  _E  ,  On ,  x
)  =  x )
87reseq2d 4971 . . . . . 6  |-  ( x  e.  On  ->  ( B  |`  Pred (  _E  ,  On ,  x )
)  =  ( B  |`  x ) )
98fveq2d 5545 . . . . 5  |-  ( x  e.  On  ->  ( G `  ( B  |` 
Pred (  _E  ,  On ,  x )
) )  =  ( G `  ( B  |`  x ) ) )
109eqeq2d 2307 . . . 4  |-  ( x  e.  On  ->  (
( B `  x
)  =  ( G `
 ( B  |`  Pred (  _E  ,  On ,  x ) ) )  <-> 
( B `  x
)  =  ( G `
 ( B  |`  x ) ) ) )
1110ralbiia 2588 . . 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 4591 . . . 4  |-  _E  We  On
14 epse 4392 . . . 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 24347 . . . . 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 2316 . . . 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 24346 . . 3  |-  ( ( B  Fn  On  /\  A. w  e.  On  ( B `  w )  =  ( G `  ( B  |`  Pred (  _E  ,  On ,  w
) ) ) )  ->  F  =  B )
2019eqcomd 2301 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843    _E cep 4319   Oncon0 4408    |` cres 4707    Fn wfn 5266   ` cfv 5271   Predcpred 24238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pred 24239
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