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Theorem rdgprc0 29443
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 4940 . . . 4  |-  (/)  e.  On
2 rdgval 7104 . . . 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 5288 . . . 4  |-  ( rec ( F ,  I
)  |`  (/) )  =  (/)
54fveq2i 5875 . . 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 2486 . 2  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
7 eqeq1 2461 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =  (/)  <->  (/)  =  (/) ) )
8 dmeq 5213 . . . . . . . . . 10  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
9 limeq 4899 . . . . . . . . . 10  |-  ( dom  g  =  dom  (/)  ->  ( Lim  dom  g  <->  Lim  dom  (/) ) )
108, 9syl 16 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( Lim 
dom  g  <->  Lim  dom  (/) ) )
11 rneq 5238 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ran  g  =  ran  (/) )
1211unieqd 4261 . . . . . . . . 9  |-  ( g  =  (/)  ->  U. ran  g  =  U. ran  (/) )
13 id 22 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  g  =  (/) )
148unieqd 4261 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  U. dom  g  =  U. dom  (/) )
1513, 14fveq12d 5878 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( g `
 U. dom  g
)  =  ( (/) ` 
U. dom  (/) ) )
1615fveq2d 5876 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F `
 ( g `  U. dom  g ) )  =  ( F `  ( (/) `  U. dom  (/) ) ) )
1710, 12, 16ifbieq12d 3971 . . . . . . . 8  |-  ( g  =  (/)  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )
187, 17ifbieq2d 3969 . . . . . . 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 2526 . . . . . 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 2457 . . . . . . 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 5508 . . . . . 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 3259 . . . . 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 2457 . . . . . . . . 9  |-  (/)  =  (/)
2423iftruei 3951 . . . . . . . 8  |-  if (
(/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  =  I
2524eleq1i 2534 . . . . . . 7  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  <->  I  e.  _V )
2625biimpi 194 . . . . . 6  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  ->  I  e.  _V )
2726adantl 466 . . . . 5  |-  ( (
(/)  e.  _V  /\  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V )  ->  I  e.  _V )
2822, 27sylbi 195 . . . 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 135 . . 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 5896 . . 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 16 . 2  |-  ( -.  I  e.  _V  ->  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  (/) )  =  (/) )
326, 31syl5eq 2510 1  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ifcif 3944   U.cuni 4251    |-> cmpt 4515   Oncon0 4887   Lim wlim 4888   dom cdm 5008   ran crn 5009    |` cres 5010   ` cfv 5594   reccrdg 7093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-recs 7060  df-rdg 7094
This theorem is referenced by:  rdgprc  29444
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