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Theorem tfr3 6383
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
StepHypRef Expression
1 nfv 1629 . . . 4  |-  F/ x  B  Fn  On
2 nfra1 2568 . . . 4  |-  F/ x A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )
31, 2nfan 1737 . . 3  |-  F/ x
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
4 nfv 1629 . . . . . 6  |-  F/ x
( B `  y
)  =  ( F `
 y )
53, 4nfim 1735 . . . . 5  |-  F/ x
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) )
6 fveq2 5458 . . . . . . 7  |-  ( x  =  y  ->  ( B `  x )  =  ( B `  y ) )
7 fveq2 5458 . . . . . . 7  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
86, 7eqeq12d 2272 . . . . . 6  |-  ( x  =  y  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( B `  y )  =  ( F `  y ) ) )
98imbi2d 309 . . . . 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 2605 . . . . . 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 ra4 2578 . . . . . . . . . 10  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )  ->  ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) ) )
12 onss 4554 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  On  ->  x  C_  On )
13 tfr.1 . . . . . . . . . . . . . . . . . . . . . 22  |-  F  = recs ( G )
1413tfr1 6381 . . . . . . . . . . . . . . . . . . . . 21  |-  F  Fn  On
15 fvreseq 5562 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( B  Fn  On  /\  F  Fn  On )  /\  x  C_  On )  ->  ( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
1614, 15mpanl2 665 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
17 fveq2 5458 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  |`  x )  =  ( F  |`  x )  ->  ( G `  ( B  |`  x ) )  =  ( G `  ( F  |`  x ) ) )
1816, 17syl6bir 222 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
1912, 18sylan2 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  Fn  On  /\  x  e.  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2019ancoms 441 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  B  Fn  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2120imp 420 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  ->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) )
2221adantr 453 . . . . . . . . . . . . . . 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 6382 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
2423jctr 528 . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) ) )
27 eqeq12 2270 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) )  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) )
2826, 27syl6 31 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) ) )
2928imp 420 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  ->  ( B `  x
)  =  ( G `
 ( B  |`  x ) ) )  /\  x  e.  On )  ->  ( ( B `
 x )  =  ( F `  x
)  <->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
3029adantl 454 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . 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 598 . . . . . . . . . . . . 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 81 . . . . . . . . . . . 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 592 . . . . . . . . . . 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 46 . . . . . . . . . 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 17 . . . . . . . . 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 77 . . . . . . . 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 422 . . . . . . 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 25 . . . . . 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 210 . . . . 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 4618 . . . 4  |-  ( x  e.  On  ->  (
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) ) )
4241com12 29 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( B `  x )  =  ( F `  x ) ) )
433, 42ralrimi 2599 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  A. x  e.  On  ( B `  x )  =  ( F `  x ) )
44 eqfnfv 5556 . . . 4  |-  ( ( B  Fn  On  /\  F  Fn  On )  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4514, 44mpan2 655 . . 3  |-  ( B  Fn  On  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4645biimpar 473 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( F `  x ) )  ->  B  =  F )
4743, 46syldan 458 1  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    C_ wss 3127   Oncon0 4364    |` cres 4663    Fn wfn 4668   ` cfv 4673  recscrecs 6355
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356
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