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Theorem tfrlemi14d 6196
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 6181 . . 3  |-  Ord  dom recs ( F )
3 ordsson 4376 . . 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 6195 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  E. g
( g  Fn  z  /\  A. u  e.  z  ( g `  u
)  =  ( F `
 ( g  |`  u ) ) ) )
75ad2antrr 477 . . . . . . . . 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 502 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  z  e.  On )
9 simprl 503 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  On )  /\  (
g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u
) ) ) )  ->  g  Fn  z
)
10 fneq2 5180 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  (
g  Fn  w  <->  g  Fn  z ) )
11 raleq 2601 . . . . . . . . . . . . 13  |-  ( w  =  z  ->  ( A. u  e.  w  ( g `  u
)  =  ( F `
 ( g  |`  u ) )  <->  A. u  e.  z  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1210, 11anbi12d 462 . . . . . . . . . . . 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 2761 . . . . . . . . . . 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 465 . . . . . . . . . 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 2661 . . . . . . . . . . 11  |-  g  e. 
_V
161, 15tfrlem3a 6173 . . . . . . . . . 10  |-  ( g  e.  A  <->  E. w  e.  On  ( g  Fn  w  /\  A. u  e.  w  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
1714, 16sylibr 133 . . . . . . . . 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 6188 . . . . . . . 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 2661 . . . . . . . . . . . 12  |-  z  e. 
_V
205tfrlem3-2d 6175 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
2120simprd 113 . . . . . . . . . . . 12  |-  ( ph  ->  ( F `  g
)  e.  _V )
22 opexg 4118 . . . . . . . . . . . 12  |-  ( ( z  e.  _V  /\  ( F `  g )  e.  _V )  ->  <. z ,  ( F `
 g ) >.  e.  _V )
2319, 21, 22sylancr 408 . . . . . . . . . . 11  |-  ( ph  -> 
<. z ,  ( F `
 g ) >.  e.  _V )
24 snidg 3522 . . . . . . . . . . 11  |-  ( <.
z ,  ( F `
 g ) >.  e.  _V  ->  <. z ,  ( F `  g
) >.  e.  { <. z ,  ( F `  g ) >. } )
25 elun2 3212 . . . . . . . . . . 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 477 . . . . . . . . 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 4712 . . . . . . . . . . 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 408 . . . . . . . . . 10  |-  ( ph  ->  ( <. z ,  ( F `  g )
>.  e.  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  z  e.  dom  ( g  u. 
{ <. z ,  ( F `  g )
>. } ) ) )
3029ad2antrr 477 . . . . . . . . 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 4707 . . . . . . . . . 10  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  dom  h  =  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) )
3332eleq2d 2185 . . . . . . . . 9  |-  ( h  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  ->  (
z  e.  dom  h  <->  z  e.  dom  ( g  u.  { <. z ,  ( F `  g ) >. } ) ) )
3433rspcev 2761 . . . . . . . 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 406 . . . . . . 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 1852 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  E. h  e.  A  z  e.  dom  h )
37 eliun 3785 . . . . . 6  |-  ( z  e.  U_ h  e.  A  dom  h  <->  E. h  e.  A  z  e.  dom  h )
3836, 37sylibr 133 . . . . 5  |-  ( (
ph  /\  z  e.  On )  ->  z  e. 
U_ h  e.  A  dom  h )
3938ex 114 . . . 4  |-  ( ph  ->  ( z  e.  On  ->  z  e.  U_ h  e.  A  dom  h ) )
4039ssrdv 3071 . . 3  |-  ( ph  ->  On  C_  U_ h  e.  A  dom  h )
411recsfval 6178 . . . . 5  |- recs ( F )  =  U. A
4241dmeqi 4708 . . . 4  |-  dom recs ( F )  =  dom  U. A
43 dmuni 4717 . . . 4  |-  dom  U. A  =  U_ h  e.  A  dom  h
4442, 43eqtri 2136 . . 3  |-  dom recs ( F )  =  U_ h  e.  A  dom  h
4540, 44sseqtrrdi 3114 . 2  |-  ( ph  ->  On  C_  dom recs ( F ) )
464, 45eqssd 3082 1  |-  ( ph  ->  dom recs ( F )  =  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   _Vcvv 2658    u. cun 3037    C_ wss 3039   {csn 3495   <.cop 3498   U.cuni 3704   U_ciun 3781   Ord word 4252   Oncon0 4253   dom cdm 4507    |` cres 4509   Fun wfun 5085    Fn wfn 5086   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfri1d  6198
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