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Theorem tfr0dm 5991
Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr0dm  |-  ( ( G `  (/) )  e.  V  ->  (/)  e.  dom  F )

Proof of Theorem tfr0dm
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3925 . . . . 5  |-  (/)  e.  _V
2 opexg 4011 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
31, 2mpan 415 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
4 snidg 3441 . . . 4  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  -> 
<. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } )
53, 4syl 14 . . 3  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  { <. (/) ,  ( G `
 (/) ) >. } )
6 fnsng 4997 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  { <. (/)
,  ( G `  (/) ) >. }  Fn  { (/)
} )
71, 6mpan 415 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  { <. (/) ,  ( G `  (/) ) >. }  Fn  { (/) } )
8 fvsng 5411 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
91, 8mpan 415 . . . . . 6  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
10 res0 4664 . . . . . . 7  |-  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) )  =  (/)
1110fveq2i 5232 . . . . . 6  |-  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) )  =  ( G `  (/) )
129, 11syl6eqr 2133 . . . . 5  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
13 fveq2 5229 . . . . . . 7  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) ) )
14 reseq2 4655 . . . . . . . 8  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
)  =  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) )
1514fveq2d 5233 . . . . . . 7  |-  ( y  =  (/)  ->  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) ) )
1613, 15eqeq12d 2097 . . . . . 6  |-  ( y  =  (/)  ->  ( ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) ) )
171, 16ralsn 3454 . . . . 5  |-  ( A. y  e.  { (/) }  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
1812, 17sylibr 132 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
19 suc0 4194 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 0elon 4175 . . . . . . 7  |-  (/)  e.  On
2120onsuci 4288 . . . . . 6  |-  suc  (/)  e.  On
2219, 21eqeltrri 2156 . . . . 5  |-  { (/) }  e.  On
23 fneq2 5039 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x 
<->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  {
(/) } ) )
24 raleq 2554 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
2523, 24anbi12d 457 . . . . . 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 2710 . . . . 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 415 . . . 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 403 . . 3  |-  ( ( G `  (/) )  e.  V  ->  E. x  e.  On  ( { <. (/)
,  ( G `  (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
29 snexg 3976 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  ->  { <. (/) ,  ( G `
 (/) ) >. }  e.  _V )
30 eleq2 2146 . . . . . . 7  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( <. (/) ,  ( G `  (/) ) >.  e.  f  <->  <. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } ) )
31 fneq1 5038 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  Fn  x  <->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  x ) )
32 fveq1 5228 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  y ) )
33 reseq1 4654 . . . . . . . . . . . 12  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  |`  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) )
3433fveq2d 5233 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( G `  ( f  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
3532, 34eqeq12d 2097 . . . . . . . . . 10  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( f `
 y )  =  ( G `  (
f  |`  y ) )  <-> 
( { <. (/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3635ralbidv 2373 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) )  <->  A. y  e.  x  ( { <.
(/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3731, 36anbi12d 457 . . . . . . . 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 2374 . . . . . . 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 457 . . . . . 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 2695 . . . . 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 2149 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  F  <->  <. (/)
,  ( G `  (/) ) >.  e. recs ( G ) )
44 df-recs 5974 . . . . . 6  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
4544eleq2i 2149 . . . . 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 3633 . . . . 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 204 . . . 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 160 . . 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 424 . 2  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  F )
50 opeldmg 4588 . . 3  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( <.
(/) ,  ( G `  (/) ) >.  e.  F  -> 
(/)  e.  dom  F ) )
511, 50mpan 415 . 2  |-  ( ( G `  (/) )  e.  V  ->  ( <. (/)
,  ( G `  (/) ) >.  e.  F  -> 
(/)  e.  dom  F ) )
5249, 51mpd 13 1  |-  ( ( G `  (/) )  e.  V  ->  (/)  e.  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610   (/)c0 3267   {csn 3416   <.cop 3419   U.cuni 3621   Oncon0 4146   suc csuc 4148   dom cdm 4391    |` cres 4393    Fn wfn 4947   ` cfv 4952  recscrecs 5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960  df-recs 5974
This theorem is referenced by:  tfr0  5992
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