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Theorem rdgprc0 27536
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 4768 . . . 4  |-  (/)  e.  On
2 rdgval 6872 . . . 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 5111 . . . 4  |-  ( rec ( F ,  I
)  |`  (/) )  =  (/)
54fveq2i 5691 . . 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 2461 . 2  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
7 eqeq1 2447 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =  (/)  <->  (/)  =  (/) ) )
8 dmeq 5036 . . . . . . . . . 10  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
9 limeq 4727 . . . . . . . . . 10  |-  ( dom  g  =  dom  (/)  ->  ( Lim  dom  g  <->  Lim  dom  (/) ) )
108, 9syl 16 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( Lim 
dom  g  <->  Lim  dom  (/) ) )
11 rneq 5061 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ran  g  =  ran  (/) )
1211unieqd 4098 . . . . . . . . 9  |-  ( g  =  (/)  ->  U. ran  g  =  U. ran  (/) )
13 id 22 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  g  =  (/) )
148unieqd 4098 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  U. dom  g  =  U. dom  (/) )
1513, 14fveq12d 5694 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( g `
 U. dom  g
)  =  ( (/) ` 
U. dom  (/) ) )
1615fveq2d 5692 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F `
 ( g `  U. dom  g ) )  =  ( F `  ( (/) `  U. dom  (/) ) ) )
1710, 12, 16ifbieq12d 3813 . . . . . . . 8  |-  ( g  =  (/)  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )
187, 17ifbieq2d 3811 . . . . . . 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 2507 . . . . . 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 2441 . . . . . . 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 5330 . . . . . 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 3116 . . . . 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 2441 . . . . . . . . 9  |-  (/)  =  (/)
2423iftruei 3795 . . . . . . . 8  |-  if (
(/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  =  I
2524eleq1i 2504 . . . . . . 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 463 . . . . 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 5711 . . 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 2485 1  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   (/)c0 3634   ifcif 3788   U.cuni 4088    e. cmpt 4347   Oncon0 4715   Lim wlim 4716   dom cdm 4836   ran crn 4837    |` cres 4838   ` cfv 5415   reccrdg 6861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-recs 6828  df-rdg 6862
This theorem is referenced by:  rdgprc  27537
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