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Theorem tfr0dm 6041
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 3941 . . . . 5  |-  (/)  e.  _V
2 opexg 4029 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
31, 2mpan 415 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
4 snidg 3456 . . . 4  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  -> 
<. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } )
53, 4syl 14 . . 3  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  { <. (/) ,  ( G `
 (/) ) >. } )
6 fnsng 5026 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  { <. (/)
,  ( G `  (/) ) >. }  Fn  { (/)
} )
71, 6mpan 415 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  { <. (/) ,  ( G `  (/) ) >. }  Fn  { (/) } )
8 fvsng 5456 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
91, 8mpan 415 . . . . . 6  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
10 res0 4685 . . . . . . 7  |-  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) )  =  (/)
1110fveq2i 5271 . . . . . 6  |-  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) )  =  ( G `  (/) )
129, 11syl6eqr 2135 . . . . 5  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
13 fveq2 5268 . . . . . . 7  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) ) )
14 reseq2 4676 . . . . . . . 8  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
)  =  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) )
1514fveq2d 5272 . . . . . . 7  |-  ( y  =  (/)  ->  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) ) )
1613, 15eqeq12d 2099 . . . . . 6  |-  ( y  =  (/)  ->  ( ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) ) )
171, 16ralsn 3469 . . . . 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 4212 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 0elon 4193 . . . . . . 7  |-  (/)  e.  On
2120onsuci 4306 . . . . . 6  |-  suc  (/)  e.  On
2219, 21eqeltrri 2158 . . . . 5  |-  { (/) }  e.  On
23 fneq2 5068 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x 
<->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  {
(/) } ) )
24 raleq 2558 . . . . . . 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 2715 . . . . 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 3993 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  ->  { <. (/) ,  ( G `
 (/) ) >. }  e.  _V )
30 eleq2 2148 . . . . . . 7  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( <. (/) ,  ( G `  (/) ) >.  e.  f  <->  <. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } ) )
31 fneq1 5067 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  Fn  x  <->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  x ) )
32 fveq1 5267 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  y ) )
33 reseq1 4675 . . . . . . . . . . . 12  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  |`  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) )
3433fveq2d 5272 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( G `  ( f  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
3532, 34eqeq12d 2099 . . . . . . . . . 10  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( f `
 y )  =  ( G `  (
f  |`  y ) )  <-> 
( { <. (/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3635ralbidv 2376 . . . . . . . . 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 2377 . . . . . . 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 2700 . . . . 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 2151 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  F  <->  <. (/)
,  ( G `  (/) ) >.  e. recs ( G ) )
44 df-recs 6024 . . . . . 6  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
4544eleq2i 2151 . . . . 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 3648 . . . . 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 4609 . . 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 1287   E.wex 1424    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2615   (/)c0 3275   {csn 3431   <.cop 3434   U.cuni 3636   Oncon0 4164   suc csuc 4166   dom cdm 4411    |` cres 4413    Fn wfn 4976   ` cfv 4981  recscrecs 6023
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-res 4423  df-iota 4946  df-fun 4983  df-fn 4984  df-fv 4989  df-recs 6024
This theorem is referenced by:  tfr0  6042
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