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Theorem tfri3 5875
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 5873). 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 1402 . . . 4  F/  Fn  On
2 nfra1 2333 . . . 4  F/  On  `  G `  |`
31, 2nfan 1439 . . 3  F/  Fn  On  On  `  G `  |`
4 nfv 1402 . . . . . 6  F/ `  F `
53, 4nfim 1446 . . . . 5  F/  Fn  On  On  `  G `  |`  `  F `
6 fveq2 5103 . . . . . . 7  `  `
7 fveq2 5103 . . . . . . 7  F `  F `
86, 7eqeq12d 2036 . . . . . 6  `  F `  `  F `
98imbi2d 219 . . . . 5  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `
 F `
10 r19.21v 2374 . . . . . 6  Fn  On  On  `  G `  |`  `  F `  Fn  On  On  `  G `  |`  `  F `
11 rsp 2347 . . . . . . . . . 10  On  `  G `  |`  On  `  G `  |`
12 onss 4169 . . . . . . . . . . . . . . . . . . 19  On  C_  On
13 tfri3.1 . . . . . . . . . . . . . . . . . . . . . 22  F recs G
14 tfri3.2 . . . . . . . . . . . . . . . . . . . . . 22  Fun 
G  G `  _V
1513, 14tfri1 5873 . . . . . . . . . . . . . . . . . . . . 21  F  Fn  On
16 fvreseq 5196 . . . . . . . . . . . . . . . . . . . . 21  Fn  On  F  Fn  On  C_  On  |`  F  |`  `  F `
1715, 16mpanl2 413 . . . . . . . . . . . . . . . . . . . 20  Fn  On  C_  On  |`  F  |`  `  F `
18 fveq2 5103 . . . . . . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . . . . . . 20  On  F `  G `  F  |`
2524jctr 298 . . . . . . . . . . . . . . . . . . 19  On  `  G `  |`  On  `  G `  |`  On  F `  G `  F  |`
26 jcab 522 . . . . . . . . . . . . . . . . . . 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 2034 . . . . . . . . . . . . . . . . . 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 4234 . . . 4  On  Fn  On  On  `  G `  |`  `  F `
4342com12 27 . . 3  Fn  On  On  `  G `  |`  On  `  F `
443, 43ralrimi 2368 . 2  Fn  On  On  `  G `  |`  On  `  F `
45 eqfnfv 5190 . . . 4  Fn  On  F  Fn  On  F  On  `  F `
4615, 45mpan2 403 . . 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 1228   wcel 1374  wral 2284   _Vcvv 2535    C_ wss 2894   Oncon0 4049    |` cres 4274   Fun wfun 4823    Fn wfn 4824   ` cfv 4829  recscrecs 5841
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842
This theorem is referenced by: (None)
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