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Theorem rdgprc0 25421
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 4634 . . . 4  |-  (/)  e.  On
2 rdgval 6678 . . . 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 8 . . 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 5150 . . . 4  |-  ( rec ( F ,  I
)  |`  (/) )  =  (/)
54fveq2i 5731 . . 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 2456 . 2  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
7 eqeq1 2442 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =  (/)  <->  (/)  =  (/) ) )
8 dmeq 5070 . . . . . . . . . 10  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
9 limeq 4593 . . . . . . . . . 10  |-  ( dom  g  =  dom  (/)  ->  ( Lim  dom  g  <->  Lim  dom  (/) ) )
108, 9syl 16 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( Lim 
dom  g  <->  Lim  dom  (/) ) )
11 rneq 5095 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ran  g  =  ran  (/) )
1211unieqd 4026 . . . . . . . . 9  |-  ( g  =  (/)  ->  U. ran  g  =  U. ran  (/) )
13 id 20 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  g  =  (/) )
148unieqd 4026 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  U. dom  g  =  U. dom  (/) )
1513, 14fveq12d 5734 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( g `
 U. dom  g
)  =  ( (/) ` 
U. dom  (/) ) )
1615fveq2d 5732 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F `
 ( g `  U. dom  g ) )  =  ( F `  ( (/) `  U. dom  (/) ) ) )
1710, 12, 16ifbieq12d 3761 . . . . . . . 8  |-  ( g  =  (/)  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )
187, 17ifbieq2d 3759 . . . . . . 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 2502 . . . . . 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 2436 . . . . . . 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 5365 . . . . . 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 3094 . . . . 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 2436 . . . . . . . . 9  |-  (/)  =  (/)
24 iftrue 3745 . . . . . . . . 9  |-  ( (/)  =  (/)  ->  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  =  I )
2523, 24ax-mp 8 . . . . . . . 8  |-  if (
(/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  =  I
2625eleq1i 2499 . . . . . . 7  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  <->  I  e.  _V )
2726biimpi 187 . . . . . 6  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  ->  I  e.  _V )
2827adantl 453 . . . . 5  |-  ( (
(/)  e.  _V  /\  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V )  ->  I  e.  _V )
2922, 28sylbi 188 . . . 4  |-  ( (/)  e.  dom  ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )  ->  I  e.  _V )
3029con3i 129 . . 3  |-  ( -.  I  e.  _V  ->  -.  (/)  e.  dom  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )
31 ndmfv 5755 . . 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
) ) ) ) ) `  (/) )  =  (/) )
3230, 31syl 16 . 2  |-  ( -.  I  e.  _V  ->  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  (/) )  =  (/) )
336, 32syl5eq 2480 1  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ifcif 3739   U.cuni 4015    e. cmpt 4266   Oncon0 4581   Lim wlim 4582   dom cdm 4878   ran crn 4879    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  rdgprc  25422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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