Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rdgprc0 Unicode version

Theorem rdgprc0 24221
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 4461 . . . 4  |-  (/)  e.  On
2 rdgval 6449 . . . 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 4975 . . . 4  |-  ( rec ( F ,  I
)  |`  (/) )  =  (/)
54fveq2i 5544 . . 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 2316 . 2  |-  ( rec ( F ,  I
) `  (/) )  =  ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) `  (/) )
7 eqeq1 2302 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =  (/)  <->  (/)  =  (/) ) )
8 dmeq 4895 . . . . . . . . . 10  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
9 limeq 4420 . . . . . . . . . 10  |-  ( dom  g  =  dom  (/)  ->  ( Lim  dom  g  <->  Lim  dom  (/) ) )
108, 9syl 15 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( Lim 
dom  g  <->  Lim  dom  (/) ) )
11 rneq 4920 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ran  g  =  ran  (/) )
1211unieqd 3854 . . . . . . . . 9  |-  ( g  =  (/)  ->  U. ran  g  =  U. ran  (/) )
13 id 19 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  g  =  (/) )
148unieqd 3854 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  U. dom  g  =  U. dom  (/) )
1513, 14fveq12d 5547 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( g `
 U. dom  g
)  =  ( (/) ` 
U. dom  (/) ) )
1615fveq2d 5545 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F `
 ( g `  U. dom  g ) )  =  ( F `  ( (/) `  U. dom  (/) ) ) )
1710, 12, 16ifbieq12d 3600 . . . . . . . 8  |-  ( g  =  (/)  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )
187, 17ifbieq2d 3598 . . . . . . 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 2362 . . . . . 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 2296 . . . . . . 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 5184 . . . . . 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 2938 . . . . 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 2296 . . . . . . . . 9  |-  (/)  =  (/)
24 iftrue 3584 . . . . . . . . 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 2359 . . . . . . 7  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  <->  I  e.  _V )
2726biimpi 186 . . . . . 6  |-  ( if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V  ->  I  e.  _V )
2827adantl 452 . . . . 5  |-  ( (
(/)  e.  _V  /\  if ( (/)  =  (/) ,  I ,  if ( Lim  dom  (/)
,  U. ran  (/) ,  ( F `  ( (/) ` 
U. dom  (/) ) ) ) )  e.  _V )  ->  I  e.  _V )
2922, 28sylbi 187 . . . 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 127 . . 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 5568 . . 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 15 . 2  |-  ( -.  I  e.  _V  ->  ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  (/) )  =  (/) )
336, 32syl5eq 2340 1  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ran crn 4706    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  rdgprc  24222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
  Copyright terms: Public domain W3C validator