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Theorem tfr0dm 6185
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 4023 . . . . 5  |-  (/)  e.  _V
2 opexg 4118 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
31, 2mpan 418 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  _V )
4 snidg 3522 . . . 4  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  -> 
<. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } )
53, 4syl 14 . . 3  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  { <. (/) ,  ( G `
 (/) ) >. } )
6 fnsng 5138 . . . . 5  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  { <. (/)
,  ( G `  (/) ) >. }  Fn  { (/)
} )
71, 6mpan 418 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  { <. (/) ,  ( G `  (/) ) >. }  Fn  { (/) } )
8 fvsng 5582 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
91, 8mpan 418 . . . . . 6  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 (/) ) )
10 res0 4791 . . . . . . 7  |-  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) )  =  (/)
1110fveq2i 5390 . . . . . 6  |-  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) )  =  ( G `  (/) )
129, 11syl6eqr 2166 . . . . 5  |-  ( ( G `  (/) )  e.  V  ->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
13 fveq2 5387 . . . . . . 7  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  (/) ) )
14 reseq2 4782 . . . . . . . 8  |-  ( y  =  (/)  ->  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
)  =  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) )
1514fveq2d 5391 . . . . . . 7  |-  ( y  =  (/)  ->  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  (/) ) ) )
1613, 15eqeq12d 2130 . . . . . 6  |-  ( y  =  (/)  ->  ( ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) ) )
171, 16ralsn 3535 . . . . 5  |-  ( A. y  e.  { (/) }  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  ( { <.
(/) ,  ( G `  (/) ) >. } `  (/) )  =  ( G `
 ( { <. (/)
,  ( G `  (/) ) >. }  |`  (/) ) ) )
1812, 17sylibr 133 . . . 4  |-  ( ( G `  (/) )  e.  V  ->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
19 suc0 4301 . . . . . 6  |-  suc  (/)  =  { (/)
}
20 0elon 4282 . . . . . . 7  |-  (/)  e.  On
2120onsuci 4400 . . . . . 6  |-  suc  (/)  e.  On
2219, 21eqeltrri 2189 . . . . 5  |-  { (/) }  e.  On
23 fneq2 5180 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( { <. (/) ,  ( G `
 (/) ) >. }  Fn  x 
<->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  {
(/) } ) )
24 raleq 2601 . . . . . . 7  |-  ( x  =  { (/) }  ->  ( A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) )  <->  A. y  e.  { (/) }  ( {
<. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
2523, 24anbi12d 462 . . . . . 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 2761 . . . . 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 418 . . . 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 406 . . 3  |-  ( ( G `  (/) )  e.  V  ->  E. x  e.  On  ( { <. (/)
,  ( G `  (/) ) >. }  Fn  x  /\  A. y  e.  x  ( { <. (/) ,  ( G `
 (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
29 snexg 4076 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  _V  ->  { <. (/) ,  ( G `
 (/) ) >. }  e.  _V )
30 eleq2 2179 . . . . . . 7  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( <. (/) ,  ( G `  (/) ) >.  e.  f  <->  <. (/) ,  ( G `
 (/) ) >.  e.  { <.
(/) ,  ( G `  (/) ) >. } ) )
31 fneq1 5179 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  Fn  x  <->  { <. (/) ,  ( G `
 (/) ) >. }  Fn  x ) )
32 fveq1 5386 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f `  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. } `  y ) )
33 reseq1 4781 . . . . . . . . . . . 12  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( f  |`  y )  =  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) )
3433fveq2d 5391 . . . . . . . . . . 11  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( G `  ( f  |`  y
) )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) )
3532, 34eqeq12d 2130 . . . . . . . . . 10  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( ( f `
 y )  =  ( G `  (
f  |`  y ) )  <-> 
( { <. (/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( { <. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3635ralbidv 2412 . . . . . . . . 9  |-  ( f  =  { <. (/) ,  ( G `  (/) ) >. }  ->  ( A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) )  <->  A. y  e.  x  ( { <.
(/) ,  ( G `  (/) ) >. } `  y )  =  ( G `  ( {
<. (/) ,  ( G `
 (/) ) >. }  |`  y
) ) ) )
3731, 36anbi12d 462 . . . . . . . 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 2413 . . . . . . 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 462 . . . . . 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 2746 . . . . 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 2182 . . . . 5  |-  ( <. (/)
,  ( G `  (/) ) >.  e.  F  <->  <. (/)
,  ( G `  (/) ) >.  e. recs ( G ) )
44 df-recs 6168 . . . . . 6  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
4544eleq2i 2182 . . . . 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 3716 . . . . 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 205 . . . 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 161 . . 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 427 . 2  |-  ( ( G `  (/) )  e.  V  ->  <. (/) ,  ( G `  (/) ) >.  e.  F )
50 opeldmg 4712 . . 3  |-  ( (
(/)  e.  _V  /\  ( G `  (/) )  e.  V )  ->  ( <.
(/) ,  ( G `  (/) ) >.  e.  F  -> 
(/)  e.  dom  F ) )
511, 50mpan 418 . 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 103    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   _Vcvv 2658   (/)c0 3331   {csn 3495   <.cop 3498   U.cuni 3704   Oncon0 4253   suc csuc 4255   dom cdm 4507    |` cres 4509    Fn wfn 5086   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfr0  6186
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