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Theorem tfri3 6086
Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 6084). 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 Jim Kingdon, 4-May-2019.)
Hypotheses
Ref Expression
tfri3.1  |-  F  = recs ( G )
tfri3.2  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
Assertion
Ref Expression
tfri3  |-  ( ( 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 tfri3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfv 1464 . . . 4  |-  F/ x  B  Fn  On
2 nfra1 2405 . . . 4  |-  F/ x A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )
31, 2nfan 1500 . . 3  |-  F/ x
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )
4 nfv 1464 . . . . . 6  |-  F/ x
( B `  y
)  =  ( F `
 y )
53, 4nfim 1507 . . . . 5  |-  F/ x
( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  y )  =  ( F `  y ) )
6 fveq2 5268 . . . . . . 7  |-  ( x  =  y  ->  ( B `  x )  =  ( B `  y ) )
7 fveq2 5268 . . . . . . 7  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
86, 7eqeq12d 2099 . . . . . 6  |-  ( x  =  y  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( B `  y )  =  ( F `  y ) ) )
98imbi2d 228 . . . . 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 2446 . . . . . 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 2419 . . . . . . . . . 10  |-  ( A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) )  ->  ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) ) )
12 onss 4283 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  On  ->  x  C_  On )
13 tfri3.1 . . . . . . . . . . . . . . . . . . . . . 22  |-  F  = recs ( G )
14 tfri3.2 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
1513, 14tfri1 6084 . . . . . . . . . . . . . . . . . . . . 21  |-  F  Fn  On
16 fvreseq 5366 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( B  Fn  On  /\  F  Fn  On )  /\  x  C_  On )  ->  ( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
1715, 16mpanl2 426 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( ( B  |`  x )  =  ( F  |`  x )  <->  A. y  e.  x  ( B `  y )  =  ( F `  y ) ) )
18 fveq2 5268 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  |`  x )  =  ( F  |`  x )  ->  ( G `  ( B  |`  x ) )  =  ( G `  ( F  |`  x ) ) )
1917, 18syl6bir 162 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  Fn  On  /\  x  C_  On )  -> 
( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2012, 19sylan2 280 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  Fn  On  /\  x  e.  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2120ancoms 264 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  B  Fn  On )  ->  ( A. y  e.  x  ( B `  y )  =  ( F `  y )  ->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
2221imp 122 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  /\  B  Fn  On )  /\  A. y  e.  x  ( B `  y )  =  ( F `  y ) )  ->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) )
2322adantr 270 . . . . . . . . . . . . . . 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 ) ) )
2413, 14tfri2 6085 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  ( F `  x )  =  ( G `  ( F  |`  x ) ) )
2524jctr 308 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
26 jcab 568 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
2725, 26sylibr 132 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) ) ) )
28 eqeq12 2097 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  x
)  =  ( G `
 ( B  |`  x ) )  /\  ( F `  x )  =  ( G `  ( F  |`  x ) ) )  ->  (
( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) )
2927, 28syl6 33 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  ->  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( ( B `  x
)  =  ( F `
 x )  <->  ( G `  ( B  |`  x
) )  =  ( G `  ( F  |`  x ) ) ) ) )
3029imp 122 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  ->  ( B `  x
)  =  ( G `
 ( B  |`  x ) ) )  /\  x  e.  On )  ->  ( ( B `
 x )  =  ( F `  x
)  <->  ( G `  ( B  |`  x ) )  =  ( G `
 ( F  |`  x ) ) ) )
3130adantl 271 . . . . . . . . . . . . . . 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 ) ) ) )
3223, 31mpbird 165 . . . . . . . . . . . . . 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 ) )
3332exp43 364 . . . . . . . . . . . . 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 ) ) ) ) )
3433com4t 84 . . . . . . . . . . . 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 ) ) ) ) )
3534exp4a 358 . . . . . . . . . . 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 ) ) ) ) ) )
3635pm2.43d 49 . . . . . . . . . 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 ) ) ) ) )
3711, 36syl 14 . . . . . . . . 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 ) ) ) ) )
3837com3l 80 . . . . . . . 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 ) ) ) ) )
3938impd 251 . . . . . . 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 ) ) ) )
4039a2d 26 . . . . . 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 ) ) ) )
4110, 40syl5bi 150 . . . . 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 ) ) ) )
425, 9, 41tfis2f 4372 . . . 4  |-  ( x  e.  On  ->  (
( B  Fn  On  /\ 
A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  ( B `  x )  =  ( F `  x ) ) )
4342com12 30 . . 3  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  (
x  e.  On  ->  ( B `  x )  =  ( F `  x ) ) )
443, 43ralrimi 2440 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  A. x  e.  On  ( B `  x )  =  ( F `  x ) )
45 eqfnfv 5360 . . . 4  |-  ( ( B  Fn  On  /\  F  Fn  On )  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4615, 45mpan2 416 . . 3  |-  ( B  Fn  On  ->  ( B  =  F  <->  A. x  e.  On  ( B `  x )  =  ( F `  x ) ) )
4746biimpar 291 . 2  |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( F `  x ) )  ->  B  =  F )
4844, 47syldan 276 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 102    <-> wb 103    = wceq 1287    e. wcel 1436   A.wral 2355   _Vcvv 2615    C_ wss 2988   Oncon0 4164    |` cres 4413   Fun wfun 4975    Fn wfn 4976   ` cfv 4981  recscrecs 6023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-recs 6024
This theorem is referenced by: (None)
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