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Theorem tfrlemi14d 5978
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 5965 . . 3  |-  Ord  dom recs ( F )
3 ordsson 4246 . . 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 5977 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
75ad2antrr 465 . . . . . . . . 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 490 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  z  e.  On )
9 simprl 491 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  g  Fn  z
)
10 fneq2 5016 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  (
g  Fn  w  <->  g  Fn  z ) )
11 raleq 2522 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  ( A. u  e.  w  ( g `  u
)  =  ( F `
 ( g  |`  u ) )  <->  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1210, 11anbi12d 450 . . . . . . . . . . . 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 2673 . . . . . . . . . . 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 453 . . . . . . . . . 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 2577 . . . . . . . . . . 11  |-  g  e. 
_V
161, 15tfrlem3a 5956 . . . . . . . . . 10  |-  ( g  e.  A  <->  E. w  e.  On  ( g  Fn  w  /\  A. u  e.  w  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1714, 16sylibr 141 . . . . . . . . 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 5970 . . . . . . . 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 2577 . . . . . . . . . . . 12  |-  z  e. 
_V
205tfrlem3-2d 5959 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
2120simprd 111 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  g
)  e.  _V )
22 opexg 3992 . . . . . . . . . . . 12  |-  ( ( z  e.  _V  /\  ( F `  g )  e.  _V )  ->  <. z ,  ( F `
 g ) >.  e.  _V )
2319, 21, 22sylancr 399 . . . . . . . . . . 11  |-  ( ph  -> 
<. z ,  ( F `
 g ) >.  e.  _V )
24 snidg 3428 . . . . . . . . . . 11  |-  ( <.
z ,  ( F `
 g ) >.  e.  _V  ->  <. z ,  ( F `  g
) >.  e.  { <. z ,  ( F `  g ) >. } )
25 elun2 3139 . . . . . . . . . . 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 465 . . . . . . . . 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 4568 . . . . . . . . . . 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 399 . . . . . . . . . 10  |-  ( ph  ->  ( <. z ,  ( F `  g )
>.  e.  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  z  e.  dom  ( g  u. 
{ <. z ,  ( F `  g )
>. } ) ) )
3029ad2antrr 465 . . . . . . . . 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 4563 . . . . . . . . . 10  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  dom  h  =  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) )
3332eleq2d 2123 . . . . . . . . 9  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  (
z  e.  dom  h  <->  z  e.  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) ) )
3433rspcev 2673 . . . . . . . 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 397 . . . . . . 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 1794 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  E. h  e.  A  z  e.  dom  h )
37 eliun 3689 . . . . . 6  |-  ( z  e.  U_ h  e.  A  dom  h  <->  E. h  e.  A  z  e.  dom  h )
3836, 37sylibr 141 . . . . 5  |-  ( (
ph  /\  z  e.  On )  ->  z  e. 
U_ h  e.  A  dom  h )
3938ex 112 . . . 4  |-  ( ph  ->  ( z  e.  On  ->  z  e.  U_ h  e.  A  dom  h ) )
4039ssrdv 2979 . . 3  |-  ( ph  ->  On  C_  U_ h  e.  A  dom  h )
411recsfval 5962 . . . . 5  |- recs ( F )  =  U. A
4241dmeqi 4564 . . . 4  |-  dom recs ( F )  =  dom  U. A
43 dmuni 4573 . . . 4  |-  dom  U. A  =  U_ h  e.  A  dom  h
4442, 43eqtri 2076 . . 3  |-  dom recs ( F )  =  U_ h  e.  A  dom  h
4540, 44syl6sseqr 3020 . 2  |-  ( ph  ->  On  C_  dom recs ( F ) )
464, 45eqssd 2990 1  |-  ( ph  ->  dom recs ( F )  =  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   _Vcvv 2574    u. cun 2943    C_ wss 2945   {csn 3403   <.cop 3406   U.cuni 3608   U_ciun 3685   Ord word 4127   Oncon0 4128   dom cdm 4373    |` cres 4375   Fun wfun 4924    Fn wfn 4925   ` cfv 4930  recscrecs 5950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951
This theorem is referenced by:  tfri1d  5980
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