ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr0 Unicode version

Theorem tfr0 5937
Description: Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr0  |-  ( ( G `  (/) )  e.  V  ->  ( F `  (/) )  =  ( G `  (/) ) )

Proof of Theorem tfr0
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3884 . . . . 5  |-  (/)  e.  _V
2 opexg 3964 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
31, 2mpan 400 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
4 snidg 3400 . . . 4  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  -> 
<. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } )
53, 4syl 14 . . 3  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  { <. (/) ,  ( G `
 (/) ) >. } )
6 fnsng 4947 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  { <. (/)
,  ( G `  (/) ) >. }  Fn  { (/)
} )
71, 6mpan 400 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  { <. (/) ,  ( G `  (/) ) >. }  Fn  { (/) } )
8 fvsng 5359 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
91, 8mpan 400 . . . . . 6  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
10 res0 4616 . . . . . . 7  |-  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) )  =  (/)
1110fveq2i 5181 . . . . . 6  |-  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) )  =  ( G `  (/) )
129, 11syl6eqr 2090 . . . . 5  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
13 fveq2 5178 . . . . . . 7  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) ) )
14 reseq2 4607 . . . . . . . 8  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
)  =  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) )
1514fveq2d 5182 . . . . . . 7  |-  ( y  =  (/)  ->  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) ) )
1613, 15eqeq12d 2054 . . . . . 6  |-  ( y  =  (/)  ->  ( ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) ) )
171, 16ralsn 3414 . . . . 5  |-  ( A. y  e.  { (/) }  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
1812, 17sylibr 137 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
19 suc0 4148 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 0elon 4129 . . . . . . 7  |-  (/)  e.  On
2120onsuci 4242 . . . . . 6  |-  suc  (/)  e.  On
2219, 21eqeltrri 2111 . . . . 5  |-  { (/) }  e.  On
23 fneq2 4988 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x 
<->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  {
(/) } ) )
24 raleq 2505 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
2523, 24anbi12d 442 . . . . . 6  |-  ( x  =  { (/) }  ->  ( ( { <. (/) ,  ( G `  (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. }  Fn  {
(/) }  /\  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) ) )
2625rspcev 2656 . . . . 5  |-  ( ( { (/) }  e.  On  /\  ( { <. (/) ,  ( G `  (/) ) >. }  Fn  { (/) }  /\  A. y  e.  { (/) }  ( { <. (/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )  ->  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) )
2722, 26mpan 400 . . . 4  |-  ( ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  {
(/) }  /\  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )  ->  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) )
287, 18, 27syl2anc 391 . . 3  |-  ( ( G `  (/) )  e.  V  ->  E. x  e.  On  ( { <. (/)
,  ( G `  (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
29 snexg 3936 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  ->  { <. (/) ,  ( G `
 (/) ) >. }  e.  _V )
30 eleq2 2101 . . . . . . 7  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( <. (/) ,  ( G `  (/) ) >.  e.  f  <->  <. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } ) )
31 fneq1 4987 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  Fn  x  <->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  x ) )
32 fveq1 5177 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  y ) )
33 reseq1 4606 . . . . . . . . . . . 12  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  |`  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) )
3433fveq2d 5182 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( G `  ( f  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
3532, 34eqeq12d 2054 . . . . . . . . . 10  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( f `
 y )  =  ( G `  (
f  |`  y ) )  <-> 
( { <. (/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3635ralbidv 2326 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) )  <->  A. y  e.  x  ( { <.
(/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3731, 36anbi12d 442 . . . . . . . 8  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) ) )
3837rexbidv 2327 . . . . . . 7  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )  <->  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) ) )
3930, 38anbi12d 442 . . . . . 6  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( <. (/)
,  ( G `  (/) ) >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) )  <->  ( <. (/) ,  ( G `  (/) ) >.  e.  { <. (/) ,  ( G `
 (/) ) >. }  /\  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) ) ) )
4039spcegv 2641 . . . . 5  |-  ( {
<. (/) ,  ( G `
 (/) ) >. }  e.  _V  ->  ( ( <. (/)
,  ( G `  (/) ) >.  e.  { <. (/)
,  ( G `  (/) ) >. }  /\  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) )  ->  E. f ( <. (/)
,  ( G `  (/) ) >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) ) ) )
413, 29, 403syl 17 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  ( ( <.
(/) ,  ( G `  (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. }  /\  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) )  ->  E. f ( <. (/)
,  ( G `  (/) ) >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) ) ) )
42 tfr.1 . . . . . 6  |-  F  = recs ( G )
4342eleq2i 2104 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  F  <->  <. (/)
,  ( G `  (/) ) >.  e. recs ( G ) )
44 df-recs 5920 . . . . . 6  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
4544eleq2i 2104 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e. recs ( G )  <->  <. (/) ,  ( G `
 (/) ) >.  e.  U. { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) } )
46 eluniab 3592 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  <->  E. f ( <. (/)
,  ( G `  (/) ) >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) ) )
4743, 45, 463bitri 195 . . . 4  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  F  <->  E. f ( <. (/) ,  ( G `  (/) ) >.  e.  f  /\  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) ) )
4841, 47syl6ibr 151 . . 3  |-  ( ( G `  (/) )  e.  V  ->  ( ( <.
(/) ,  ( G `  (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. }  /\  E. x  e.  On  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/)
,  ( G `  (/) ) >. } `  y
)  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) ) ) )  ->  <. (/) ,  ( G `
 (/) ) >.  e.  F
) )
495, 28, 48mp2and 409 . 2  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  F )
50 opeldmg 4540 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( <.
(/) ,  ( G `  (/) ) >.  e.  F  -> 
(/)  e.  dom  F ) )
511, 50mpan 400 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  ( <. (/)
,  ( G `  (/) ) >.  e.  F  -> 
(/)  e.  dom  F ) )
5242tfr2a 5936 . . . 4  |-  ( (/)  e.  dom  F  ->  ( F `  (/) )  =  ( G `  ( F  |`  (/) ) ) )
5351, 52syl6 29 . . 3  |-  ( ( G `  (/) )  e.  V  ->  ( <. (/)
,  ( G `  (/) ) >.  e.  F  ->  ( F `  (/) )  =  ( G `  ( F  |`  (/) ) ) ) )
54 res0 4616 . . . . 5  |-  ( F  |`  (/) )  =  (/)
5554fveq2i 5181 . . . 4  |-  ( G `
 ( F  |`  (/) ) )  =  ( G `  (/) )
5655eqeq2i 2050 . . 3  |-  ( ( F `  (/) )  =  ( G `  ( F  |`  (/) ) )  <->  ( F `  (/) )  =  ( G `  (/) ) )
5753, 56syl6ib 150 . 2  |-  ( ( G `  (/) )  e.  V  ->  ( <. (/)
,  ( G `  (/) ) >.  e.  F  ->  ( F `  (/) )  =  ( G `  (/) ) ) )
5849, 57mpd 13 1  |-  ( ( G `  (/) )  e.  V  ->  ( F `  (/) )  =  ( G `  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557   (/)c0 3224   {csn 3375   <.cop 3378   U.cuni 3580   Oncon0 4100   suc csuc 4102   dom cdm 4345    |` cres 4347    Fn wfn 4897   ` cfv 4902  recscrecs 5919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920
This theorem is referenced by:  rdg0  5974  frec0g  5983
  Copyright terms: Public domain W3C validator