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Theorem tfr3 6415
Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally we show that  F is unique. We do this by showing that any class  B with the same properties of  F that we showed in parts 1 and 2 is identical to  F. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Distinct variable groups:    x, B    x, F    x, G

Proof of Theorem tfr3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . . 4  |-  F/ x  B  Fn  On
2 nfra1 2593 . . . 4  |-  F/ x A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )
31, 2nfan 1771 . . 3  |-  F/ x
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
4 nfv 1605 . . . . . 6  |-  F/ x
( B `  y
)  =  ( F `
 y )
53, 4nfim 1769 . . . . 5  |-  F/ x
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) )
6 fveq2 5525 . . . . . . 7  |-  ( x  =  y  ->  ( B `  x )  =  ( B `  y ) )
7 fveq2 5525 . . . . . . 7  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
86, 7eqeq12d 2297 . . . . . 6  |-  ( x  =  y  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( B `  y )  =  ( F `  y ) ) )
98imbi2d 307 . . . . 5  |-  ( x  =  y  ->  (
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) )  <->  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) ) ) )
10 r19.21v 2630 . . . . . 6  |-  ( A. y  e.  x  (
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) )  <->  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
11 rsp 2603 . . . . . . . . . 10  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )  ->  ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) ) )
12 onss 4582 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  On  ->  x  C_  On )
13 tfr.1 . . . . . . . . . . . . . . . . . . . . . 22  |-  F  = recs ( G )
1413tfr1 6413 . . . . . . . . . . . . . . . . . . . . 21  |-  F  Fn  On
15 fvreseq 5628 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( B  Fn  On  /\  F  Fn  On )  /\  x  C_  On )  ->  ( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
1614, 15mpanl2 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
17 fveq2 5525 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  |`  x )  =  ( F  |`  x )  ->  ( G `  ( B  |`  x ) )  =  ( G `  ( F  |`  x ) ) )
1816, 17syl6bir 220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
1912, 18sylan2 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  Fn  On  /\  x  e.  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2019ancoms 439 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  B  Fn  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2120imp 418 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  ->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) )
2221adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  /\  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  /\  x  e.  On ) )  -> 
( G `  ( B  |`  x ) )  =  ( G `  ( F  |`  x ) ) )
2313tfr2 6414 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
2423jctr 526 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
( x  e.  On  ->  ( B `  x
)  =  ( G `
 ( B  |`  x ) ) )  /\  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) ) )
25 jcab 833 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  On  ->  ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) )  <->  ( (
x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  /\  (
x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) ) )
2624, 25sylibr 203 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) ) )
27 eqeq12 2295 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) )  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) )
2826, 27syl6 29 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) ) )
2928imp 418 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  ->  ( B `  x
)  =  ( G `
 ( B  |`  x ) ) )  /\  x  e.  On )  ->  ( ( B `
 x )  =  ( F `  x
)  <->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
3029adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  /\  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  /\  x  e.  On ) )  -> 
( ( B `  x )  =  ( F `  x )  <-> 
( G `  ( B  |`  x ) )  =  ( G `  ( F  |`  x ) ) ) )
3122, 30mpbird 223 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  /\  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  /\  x  e.  On ) )  -> 
( B `  x
)  =  ( F `
 x ) )
3231exp43 595 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  B  Fn  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( B `  x )  =  ( F `  x ) ) ) ) )
3332com4t 79 . . . . . . . . . . . 12  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( x  e.  On  /\  B  Fn  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) ) )
3433exp4a 589 . . . . . . . . . . 11  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( x  e.  On  ->  ( B  Fn  On  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) ) ) )
3534pm2.43d 44 . . . . . . . . . 10  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( B  Fn  On  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) ) )
3611, 35syl 15 . . . . . . . . 9  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )  ->  ( x  e.  On  ->  ( B  Fn  On  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) ) )
3736com3l 75 . . . . . . . 8  |-  ( x  e.  On  ->  ( B  Fn  On  ->  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) ) )
3837imp3a 420 . . . . . . 7  |-  ( x  e.  On  ->  (
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( B `  x )  =  ( F `  x ) ) ) )
3938a2d 23 . . . . . 6  |-  ( x  e.  On  ->  (
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  -> 
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) ) ) )
4010, 39syl5bi 208 . . . . 5  |-  ( x  e.  On  ->  ( A. y  e.  x  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) )  ->  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) ) ) )
415, 9, 40tfis2f 4646 . . . 4  |-  ( x  e.  On  ->  (
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) ) )
4241com12 27 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( B `  x )  =  ( F `  x ) ) )
433, 42ralrimi 2624 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  A. x  e.  On  ( B `  x )  =  ( F `  x ) )
44 eqfnfv 5622 . . . 4  |-  ( ( B  Fn  On  /\  F  Fn  On )  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4514, 44mpan2 652 . . 3  |-  ( B  Fn  On  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4645biimpar 471 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( F `  x ) )  ->  B  =  F )
4743, 46syldan 456 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   Oncon0 4392    |` cres 4691    Fn wfn 5250   ` cfv 5255  recscrecs 6387
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-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-we 4354  df-ord 4395  df-on 4396  df-suc 4398  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-recs 6388
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