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Theorem tfrlemi14d 6098
Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
Hypotheses
Ref Expression
tfrlemi14d.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlemi14d.2  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
Assertion
Ref Expression
tfrlemi14d  |-  ( ph  ->  dom recs ( F )  =  On )
Distinct variable groups:    x, f, y, A    f, F, x, y    ph, f, y
Allowed substitution hint:    ph( x)

Proof of Theorem tfrlemi14d
Dummy variables  g  h  u  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlemi14d.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem8 6083 . . 3  |-  Ord  dom recs ( F )
3 ordsson 4309 . . 3  |-  ( Ord 
dom recs ( F )  ->  dom recs ( F )  C_  On )
42, 3mp1i 10 . 2  |-  ( ph  ->  dom recs ( F ) 
C_  On )
5 tfrlemi14d.2 . . . . . . . 8  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
61, 5tfrlemi1 6097 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
75ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
8 simplr 497 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  z  e.  On )
9 simprl 498 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  g  Fn  z
)
10 fneq2 5103 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  (
g  Fn  w  <->  g  Fn  z ) )
11 raleq 2562 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  ( A. u  e.  w  ( g `  u
)  =  ( F `
 ( g  |`  u ) )  <->  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1210, 11anbi12d 457 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
( g  Fn  w  /\  A. u  e.  w  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) )  <-> 
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) ) )
1312rspcev 2722 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  ( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )  ->  E. w  e.  On  ( g  Fn  w  /\  A. u  e.  w  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1413adantll 460 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  E. w  e.  On  ( g  Fn  w  /\  A. u  e.  w  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
15 vex 2622 . . . . . . . . . . 11  |-  g  e. 
_V
161, 15tfrlem3a 6075 . . . . . . . . . 10  |-  ( g  e.  A  <->  E. w  e.  On  ( g  Fn  w  /\  A. u  e.  w  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1714, 16sylibr 132 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  g  e.  A
)
181, 7, 8, 9, 17tfrlemisucaccv 6090 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  e.  A
)
19 vex 2622 . . . . . . . . . . . 12  |-  z  e. 
_V
205tfrlem3-2d 6077 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
2120simprd 112 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  g
)  e.  _V )
22 opexg 4055 . . . . . . . . . . . 12  |-  ( ( z  e.  _V  /\  ( F `  g )  e.  _V )  ->  <. z ,  ( F `
 g ) >.  e.  _V )
2319, 21, 22sylancr 405 . . . . . . . . . . 11  |-  ( ph  -> 
<. z ,  ( F `
 g ) >.  e.  _V )
24 snidg 3473 . . . . . . . . . . 11  |-  ( <.
z ,  ( F `
 g ) >.  e.  _V  ->  <. z ,  ( F `  g
) >.  e.  { <. z ,  ( F `  g ) >. } )
25 elun2 3168 . . . . . . . . . . 11  |-  ( <.
z ,  ( F `
 g ) >.  e.  { <. z ,  ( F `  g )
>. }  ->  <. z ,  ( F `  g
) >.  e.  ( g  u.  { <. z ,  ( F `  g ) >. } ) )
2623, 24, 253syl 17 . . . . . . . . . 10  |-  ( ph  -> 
<. z ,  ( F `
 g ) >.  e.  ( g  u.  { <. z ,  ( F `
 g ) >. } ) )
2726ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  <. z ,  ( F `  g )
>.  e.  ( g  u. 
{ <. z ,  ( F `  g )
>. } ) )
28 opeldmg 4641 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  ( F `  g )  e.  _V )  -> 
( <. z ,  ( F `  g )
>.  e.  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  z  e.  dom  ( g  u. 
{ <. z ,  ( F `  g )
>. } ) ) )
2919, 21, 28sylancr 405 . . . . . . . . . 10  |-  ( ph  ->  ( <. z ,  ( F `  g )
>.  e.  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  z  e.  dom  ( g  u. 
{ <. z ,  ( F `  g )
>. } ) ) )
3029ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  ( <. z ,  ( F `  g ) >.  e.  ( g  u.  { <. z ,  ( F `  g ) >. } )  ->  z  e.  dom  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) )
3127, 30mpd 13 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  z  e.  dom  ( g  u.  { <. z ,  ( F `
 g ) >. } ) )
32 dmeq 4636 . . . . . . . . . 10  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  dom  h  =  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) )
3332eleq2d 2157 . . . . . . . . 9  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  (
z  e.  dom  h  <->  z  e.  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) ) )
3433rspcev 2722 . . . . . . . 8  |-  ( ( ( g  u.  { <. z ,  ( F `
 g ) >. } )  e.  A  /\  z  e.  dom  ( g  u.  { <. z ,  ( F `
 g ) >. } ) )  ->  E. h  e.  A  z  e.  dom  h )
3518, 31, 34syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  E. h  e.  A  z  e.  dom  h )
366, 35exlimddv 1826 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  E. h  e.  A  z  e.  dom  h )
37 eliun 3734 . . . . . 6  |-  ( z  e.  U_ h  e.  A  dom  h  <->  E. h  e.  A  z  e.  dom  h )
3836, 37sylibr 132 . . . . 5  |-  ( (
ph  /\  z  e.  On )  ->  z  e. 
U_ h  e.  A  dom  h )
3938ex 113 . . . 4  |-  ( ph  ->  ( z  e.  On  ->  z  e.  U_ h  e.  A  dom  h ) )
4039ssrdv 3031 . . 3  |-  ( ph  ->  On  C_  U_ h  e.  A  dom  h )
411recsfval 6080 . . . . 5  |- recs ( F )  =  U. A
4241dmeqi 4637 . . . 4  |-  dom recs ( F )  =  dom  U. A
43 dmuni 4646 . . . 4  |-  dom  U. A  =  U_ h  e.  A  dom  h
4442, 43eqtri 2108 . . 3  |-  dom recs ( F )  =  U_ h  e.  A  dom  h
4540, 44syl6sseqr 3073 . 2  |-  ( ph  ->  On  C_  dom recs ( F ) )
464, 45eqssd 3042 1  |-  ( ph  ->  dom recs ( F )  =  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    u. cun 2997    C_ wss 2999   {csn 3446   <.cop 3449   U.cuni 3653   U_ciun 3730   Ord word 4189   Oncon0 4190   dom cdm 4438    |` cres 4440   Fun wfun 5009    Fn wfn 5010   ` cfv 5015  recscrecs 6069
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-recs 6070
This theorem is referenced by:  tfri1d  6100
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