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Theorem rdgon 5913
Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)
Hypotheses
Ref Expression
rdgon.1  F  Fn  _V
rdgon.2  On
rdgon.3  On  F `  On
Assertion
Ref Expression
rdgon  On  rec F ,  `  On
Distinct variable groups:   ,   , F   ,
Allowed substitution hint:   ()

Proof of Theorem rdgon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5121 . . . . 5  rec F ,  `  rec F ,  `
21eleq1d 2103 . . . 4  rec F ,  `  On  rec F ,  `  On
32imbi2d 219 . . 3  rec F ,  `  On  rec F ,  `  On
4 fveq2 5121 . . . . 5  rec F ,  `  rec F ,  `
54eleq1d 2103 . . . 4  rec F ,  `  On  rec F ,  `  On
65imbi2d 219 . . 3  rec F ,  `  On  rec F ,  `  On
7 r19.21v 2390 . . . 4  rec F ,  `  On  rec F ,  `  On
8 rdgon.2 . . . . . . . . 9  On
9 fvres 5141 . . . . . . . . . . . . . 14  rec F ,  |`  `
 rec F ,  `
109eleq1d 2103 . . . . . . . . . . . . 13  rec F ,  |`  `  On  rec F ,  `  On
1110adantl 262 . . . . . . . . . . . 12  rec F ,  |`  `  On  rec F ,  `  On
12 rdgon.3 . . . . . . . . . . . . . . 15  On  F `  On
13 fveq2 5121 . . . . . . . . . . . . . . . . 17  F `  F `
1413eleq1d 2103 . . . . . . . . . . . . . . . 16  F `  On  F `
 On
1514cbvralv 2527 . . . . . . . . . . . . . . 15  On  F `  On  On  F `  On
1612, 15sylib 127 . . . . . . . . . . . . . 14  On  F `  On
17 fveq2 5121 . . . . . . . . . . . . . . . 16  rec F ,  |`  `  F `  F `  rec F ,  |`  `
1817eleq1d 2103 . . . . . . . . . . . . . . 15  rec F ,  |`  `  F `  On  F `
 rec F ,  |`  `  On
1918rspcv 2646 . . . . . . . . . . . . . 14  rec F ,  |`  `  On  On  F `  On  F `  rec F ,  |`  `  On
2016, 19syl5com 26 . . . . . . . . . . . . 13  rec F ,  |`  `  On  F `  rec F ,  |`  `
 On
2120adantr 261 . . . . . . . . . . . 12  rec F ,  |`  `  On  F `  rec F ,  |`  `  On
2211, 21sylbird 159 . . . . . . . . . . 11  rec F ,  `  On  F `  rec F ,  |`  `  On
2322ralimdva 2381 . . . . . . . . . 10  rec F ,  `  On  F `  rec F ,  |`  `
 On
24 vex 2554 . . . . . . . . . . 11 
_V
25 iunon 5840 . . . . . . . . . . 11  _V  F `  rec F ,  |`  `
 On  U_  F `  rec F ,  |`  `
 On
2624, 25mpan 400 . . . . . . . . . 10  F `  rec F ,  |`  `  On  U_  F `  rec F ,  |`  `
 On
2723, 26syl6 29 . . . . . . . . 9  rec F ,  `  On  U_  F `  rec F ,  |`  `
 On
28 onun2 4182 . . . . . . . . 9  On  U_  F `  rec F ,  |`  `
 On  u.  U_  F `  rec F ,  |`  `  On
298, 27, 28syl6an 1320 . . . . . . . 8  rec F ,  `  On  u.  U_  F `  rec F ,  |`  `  On
3029adantr 261 . . . . . . 7  On  rec F ,  `
 On  u.  U_  F `  rec F ,  |`  `  On
31 rdgon.1 . . . . . . . . . 10  F  Fn  _V
3231, 8jca 290 . . . . . . . . 9  F  Fn  _V  On
33 rdgivallem 5908 . . . . . . . . . 10  F  Fn  _V  On  On  rec F ,  `  u.  U_  F `  rec F ,  |`  `
34333expa 1103 . . . . . . . . 9  F  Fn  _V  On  On  rec F ,  `  u.  U_  F `  rec F ,  |`  `
3532, 34sylan 267 . . . . . . . 8  On  rec F ,  `  u.  U_  F `  rec F ,  |`  `
3635eleq1d 2103 . . . . . . 7  On  rec F ,  `  On  u.  U_  F `  rec F ,  |`  `
 On
3730, 36sylibrd 158 . . . . . 6  On  rec F ,  `
 On  rec F ,  `  On
3837expcom 109 . . . . 5  On  rec F ,  `
 On  rec F ,  `  On
3938a2d 23 . . . 4  On  rec F ,  `
 On  rec F ,  `  On
407, 39syl5bi 141 . . 3  On  rec F ,  `  On  rec F ,  `  On
413, 6, 40tfis3 4252 . 2  On  rec F ,  `  On
4241impcom 116 1  On  rec F ,  `  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    u. cun 2909   U_ciun 3648   Oncon0 4066    |` cres 4290    Fn wfn 4840   ` cfv 4845   reccrdg 5896
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-bndl 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  df-irdg 5897
This theorem is referenced by:  oacl  5979  omcl  5980  oeicl  5981
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