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Theorem rdgprc0 30511
Description: The value of the recursive definition generator at  (/) when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rdgprc0  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )

Proof of Theorem rdgprc0
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 0elon 5483 . . . 4  |-  (/)  e.  On
2 rdgval 7156 . . . 4  |-  ( (/)  e.  On  ->  ( rec ( F ,  I ) `
 (/) )  =  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  ( rec ( F ,  I
)  |`  (/) ) ) )
31, 2ax-mp 5 . . 3  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  ( rec ( F ,  I
)  |`  (/) ) )
4 res0 5115 . . . 4  |-  ( rec ( F ,  I
)  |`  (/) )  =  (/)
54fveq2i 5882 . . 3  |-  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  ( rec ( F ,  I
)  |`  (/) ) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
63, 5eqtri 2493 . 2  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
7 eqeq1 2475 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =  (/)  <->  (/)  =  (/) ) )
8 dmeq 5040 . . . . . . . . . 10  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
9 limeq 5442 . . . . . . . . . 10  |-  ( dom  g  =  dom  (/)  ->  ( Lim  dom  g  <->  Lim  dom  (/) ) )
108, 9syl 17 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( Lim 
dom  g  <->  Lim  dom  (/) ) )
11 rneq 5066 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ran  g  =  ran  (/) )
1211unieqd 4200 . . . . . . . . 9  |-  ( g  =  (/)  ->  U. ran  g  =  U. ran  (/) )
13 id 22 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  g  =  (/) )
148unieqd 4200 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  U. dom  g  =  U. dom  (/) )
1513, 14fveq12d 5885 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( g `
 U. dom  g
)  =  ( (/) ` 
U. dom  (/) ) )
1615fveq2d 5883 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F `
 ( g `  U. dom  g ) )  =  ( F `  ( (/) `  U. dom  (/) ) ) )
1710, 12, 16ifbieq12d 3899 . . . . . . . 8  |-  ( g  =  (/)  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )
187, 17ifbieq2d 3897 . . . . . . 7  |-  ( g  =  (/)  ->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )  =  if (
(/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) ) )
1918eleq1d 2533 . . . . . 6  |-  ( g  =  (/)  ->  ( if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )  e.  _V  <->  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V ) )
20 eqid 2471 . . . . . . 7  |-  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )  =  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
2120dmmpt 5337 . . . . . 6  |-  dom  (
g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )  =  { g  e.  _V  |  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )  e.  _V }
2219, 21elrab2 3186 . . . . 5  |-  ( (/)  e.  dom  ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )  <->  ( (/)  e.  _V  /\  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/) ,  U. ran  (/) ,  ( F `
 ( (/) `  U. dom  (/) ) ) ) )  e.  _V )
)
23 eqid 2471 . . . . . . . . 9  |-  (/)  =  (/)
2423iftruei 3879 . . . . . . . 8  |-  if (
(/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  =  I
2524eleq1i 2540 . . . . . . 7  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  <->  I  e.  _V )
2625biimpi 199 . . . . . 6  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  ->  I  e.  _V )
2726adantl 473 . . . . 5  |-  ( (
(/)  e.  _V  /\  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V )  ->  I  e.  _V )
2822, 27sylbi 200 . . . 4  |-  ( (/)  e.  dom  ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )  ->  I  e.  _V )
2928con3i 142 . . 3  |-  ( -.  I  e.  _V  ->  -.  (/)  e.  dom  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )
30 ndmfv 5903 . . 3  |-  ( -.  (/)  e.  dom  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )  ->  (
( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  (/) )  =  (/) )
3129, 30syl 17 . 2  |-  ( -.  I  e.  _V  ->  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  (/) )  =  (/) )
326, 31syl5eq 2517 1  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   ifcif 3872   U.cuni 4190    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   Oncon0 5430   Lim wlim 5431   ` cfv 5589   reccrdg 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-wrecs 7046  df-recs 7108  df-rdg 7146
This theorem is referenced by:  rdgprc  30512
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