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Theorem tfrcldm 6033
Description: Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f  |-  F  = recs ( G )
tfrcl.g  |-  ( ph  ->  Fun  G )
tfrcl.x  |-  ( ph  ->  Ord  X )
tfrcl.ex  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
tfrcl.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfrcl.yx  |-  ( ph  ->  Y  e.  U. X
)
Assertion
Ref Expression
tfrcldm  |-  ( ph  ->  Y  e.  dom  F
)
Distinct variable groups:    f, G, x    S, f, x    f, X, x    f, Y, x    ph, f, x
Allowed substitution hints:    F( x, f)

Proof of Theorem tfrcldm
Dummy variables  z  a  b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.yx . . 3  |-  ( ph  ->  Y  e.  U. X
)
2 eluni 3625 . . 3  |-  ( Y  e.  U. X  <->  E. z
( Y  e.  z  /\  z  e.  X
) )
31, 2sylib 120 . 2  |-  ( ph  ->  E. z ( Y  e.  z  /\  z  e.  X ) )
4 tfrcl.f . . . 4  |-  F  = recs ( G )
5 tfrcl.g . . . . 5  |-  ( ph  ->  Fun  G )
65adantr 270 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Fun  G )
7 tfrcl.x . . . . 5  |-  ( ph  ->  Ord  X )
87adantr 270 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Ord  X )
9 tfrcl.ex . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
1093adant1r 1163 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
11 feq2 5083 . . . . . . . 8  |-  ( a  =  x  ->  (
f : a --> S  <-> 
f : x --> S ) )
12 raleq 2554 . . . . . . . 8  |-  ( a  =  x  ->  ( A. b  e.  a 
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
1311, 12anbi12d 457 . . . . . . 7  |-  ( a  =  x  ->  (
( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <-> 
( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) ) )
1413cbvrexv 2583 . . . . . 6  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
15 fveq2 5230 . . . . . . . . . 10  |-  ( b  =  y  ->  (
f `  b )  =  ( f `  y ) )
16 reseq2 4656 . . . . . . . . . . 11  |-  ( b  =  y  ->  (
f  |`  b )  =  ( f  |`  y
) )
1716fveq2d 5234 . . . . . . . . . 10  |-  ( b  =  y  ->  ( G `  ( f  |`  b ) )  =  ( G `  (
f  |`  y ) ) )
1815, 17eqeq12d 2097 . . . . . . . . 9  |-  ( b  =  y  ->  (
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
1918cbvralv 2582 . . . . . . . 8  |-  ( A. b  e.  x  (
f `  b )  =  ( G `  ( f  |`  b
) )  <->  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )
2019anbi2i 445 . . . . . . 7  |-  ( ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2120rexbii 2378 . . . . . 6  |-  ( E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2214, 21bitri 182 . . . . 5  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
2322abbii 2198 . . . 4  |-  { f  |  E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  (
f `  b )  =  ( G `  ( f  |`  b
) ) ) }  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
24 tfrcl.u . . . . 5  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
2524adantlr 461 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  U. X )  ->  suc  x  e.  X )
26 simprr 499 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  e.  X )
274, 6, 8, 10, 23, 25, 26tfrcllemres 6032 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  C_  dom  F )
28 simprl 498 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  z )
2927, 28sseldd 3010 . 2  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  dom  F )
303, 29exlimddv 1821 1  |-  ( ph  ->  Y  e.  dom  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   U.cuni 3622   Ord word 4146   suc csuc 4149   dom cdm 4392    |` cres 4394   Fun wfun 4947   -->wf 4949   ` cfv 4953  recscrecs 5974
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-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309
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-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-suc 4155  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-recs 5975
This theorem is referenced by:  tfrcl  6034  frecfcllem  6074  frecsuclem  6076
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