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Theorem tfri3 5894
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 5892). Finally, we show that  F is unique. We do this by showing that any class 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 `  _V
Assertion
Ref Expression
tfri3  Fn  On  On  `  G `  |`  F
Distinct variable groups:   ,   , F   , G

Proof of Theorem tfri3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4  F/  Fn  On
2 nfra1 2349 . . . 4  F/  On  `  G `  |`
31, 2nfan 1454 . . 3  F/  Fn  On  On  `  G `  |`
4 nfv 1418 . . . . . 6  F/ `  F `
53, 4nfim 1461 . . . . 5  F/  Fn  On  On  `  G `  |`  `  F `
6 fveq2 5121 . . . . . . 7  `  `
7 fveq2 5121 . . . . . . 7  F `  F `
86, 7eqeq12d 2051 . . . . . 6  `  F `  `  F `
98imbi2d 219 . . . . 5  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `
 F `
10 r19.21v 2390 . . . . . 6  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `  F `
11 rsp 2363 . . . . . . . . . 10  On  `  G `  |`  On  `  G `  |`
12 onss 4185 . . . . . . . . . . . . . . . . . . 19  On  C_  On
13 tfri3.1 . . . . . . . . . . . . . . . . . . . . . 22  F recs G
14 tfri3.2 . . . . . . . . . . . . . . . . . . . . . 22  Fun 
G  G `  _V
1513, 14tfri1 5892 . . . . . . . . . . . . . . . . . . . . 21  F  Fn  On
16 fvreseq 5214 . . . . . . . . . . . . . . . . . . . . 21  Fn  On  F  Fn  On  C_  On  |`  F  |`  `  F `
1715, 16mpanl2 411 . . . . . . . . . . . . . . . . . . . 20  Fn  On  C_  On  |`  F  |`  `  F `
18 fveq2 5121 . . . . . . . . . . . . . . . . . . . 20  |`  F  |`  G `  |`  G `
 F  |`
1917, 18syl6bir 153 . . . . . . . . . . . . . . . . . . 19  Fn  On  C_  On  `  F `  G `  |`  G `  F  |`
2012, 19sylan2 270 . . . . . . . . . . . . . . . . . 18  Fn  On  On  `  F `  G `  |`  G `  F  |`
2120ancoms 255 . . . . . . . . . . . . . . . . 17  On  Fn  On  `  F `  G `  |`  G `  F  |`
2221imp 115 . . . . . . . . . . . . . . . 16  On  Fn  On  `  F `  G `  |`  G `  F  |`
2322adantr 261 . . . . . . . . . . . . . . 15  On  Fn  On  `  F `  On  `  G `  |`  On  G `  |`  G `  F  |`
2413, 14tfri2 5893 . . . . . . . . . . . . . . . . . . . 20  On  F `  G `  F  |`
2524jctr 298 . . . . . . . . . . . . . . . . . . 19  On  `  G `  |`  On  `  G `  |`  On  F `  G `  F  |`
26 jcab 535 . . . . . . . . . . . . . . . . . . 19  On  `  G `  |`  F `  G `  F  |`  On  `  G `  |`  On  F `  G `  F  |`
2725, 26sylibr 137 . . . . . . . . . . . . . . . . . 18  On  `  G `  |`  On  `  G `  |`  F `  G `  F  |`
28 eqeq12 2049 . . . . . . . . . . . . . . . . . 18  `  G `  |`  F `  G `  F  |`  `  F `  G `  |`  G `  F  |`
2927, 28syl6 29 . . . . . . . . . . . . . . . . 17  On  `  G `  |`  On  `  F `  G `
 |`  G `  F  |`
3029imp 115 . . . . . . . . . . . . . . . 16  On  `  G `  |`  On  `  F `  G `  |`  G `  F  |`
3130adantl 262 . . . . . . . . . . . . . . 15  On  Fn  On  `  F `  On  `  G `  |`  On  `  F `  G `
 |`  G `  F  |`
3223, 31mpbird 156 . . . . . . . . . . . . . 14  On  Fn  On  `  F `  On  `  G `  |`  On  `  F `
3332exp43 354 . . . . . . . . . . . . 13  On  Fn  On  `  F `  On  `  G `  |`  On  `  F `
3433com4t 79 . . . . . . . . . . . 12  On  `  G `  |`  On  On  Fn  On  `  F `  `  F `
3534exp4a 348 . . . . . . . . . . 11  On  `  G `  |`  On  On  Fn  On  `  F `  `  F `
3635pm2.43d 44 . . . . . . . . . 10  On  `  G `  |`  On  Fn  On  `  F `  `  F `
3711, 36syl 14 . . . . . . . . 9  On  `  G `  |`  On  Fn  On  `  F `  `  F `
3837com3l 75 . . . . . . . 8  On  Fn  On  On  `  G `  |`  `  F `  `  F `
3938impd 242 . . . . . . 7  On  Fn  On  On  `  G `  |`  `  F `  `  F `
4039a2d 23 . . . . . 6  On  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `  F `
4110, 40syl5bi 141 . . . . 5  On  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `  F `
425, 9, 41tfis2f 4250 . . . 4  On  Fn  On  On  `  G `  |`  `  F `
4342com12 27 . . 3  Fn  On  On  `  G `  |`  On  `  F `
443, 43ralrimi 2384 . 2  Fn  On  On  `  G `  |`  On  `  F `
45 eqfnfv 5208 . . . 4  Fn  On  F  Fn  On  F  On  `  F `
4615, 45mpan2 401 . . 3  Fn  On  F  On  `  F `
4746biimpar 281 . 2  Fn  On  On  `  F `  F
4844, 47syldan 266 1  Fn  On  On  `  G `  |`  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    C_ wss 2911   Oncon0 4066    |` cres 4290   Fun wfun 4839    Fn wfn 4840   ` cfv 4845  recscrecs 5860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861
This theorem is referenced by: (None)
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