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Theorem frecabex 6246
Description: The class abstraction from df-frec 6239 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
Hypotheses
Ref Expression
frecabex.sex  |-  ( ph  ->  S  e.  V )
frecabex.fvex  |-  ( ph  ->  A. y ( F `
 y )  e. 
_V )
frecabex.aex  |-  ( ph  ->  A  e.  W )
Assertion
Ref Expression
frecabex  |-  ( ph  ->  { x  |  ( E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  \/  ( dom 
S  =  (/)  /\  x  e.  A ) ) }  e.  _V )
Distinct variable groups:    x, A    x, F    x, S, y    ph, m    x, m, y    y, F
Allowed substitution hints:    ph( x, y)    A( y, m)    S( m)    F( m)    V( x, y, m)    W( x, y, m)

Proof of Theorem frecabex
StepHypRef Expression
1 omex 4465 . . . 4  |-  om  e.  _V
2 simpr 109 . . . . . . 7  |-  ( ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  ->  x  e.  ( F `  ( S `
 m ) ) )
32abssi 3136 . . . . . 6  |-  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m )
) ) }  C_  ( F `  ( S `
 m ) )
4 frecabex.sex . . . . . . . 8  |-  ( ph  ->  S  e.  V )
5 vex 2658 . . . . . . . 8  |-  m  e. 
_V
6 fvexg 5392 . . . . . . . 8  |-  ( ( S  e.  V  /\  m  e.  _V )  ->  ( S `  m
)  e.  _V )
74, 5, 6sylancl 407 . . . . . . 7  |-  ( ph  ->  ( S `  m
)  e.  _V )
8 frecabex.fvex . . . . . . 7  |-  ( ph  ->  A. y ( F `
 y )  e. 
_V )
9 fveq2 5373 . . . . . . . . 9  |-  ( y  =  ( S `  m )  ->  ( F `  y )  =  ( F `  ( S `  m ) ) )
109eleq1d 2181 . . . . . . . 8  |-  ( y  =  ( S `  m )  ->  (
( F `  y
)  e.  _V  <->  ( F `  ( S `  m
) )  e.  _V ) )
1110spcgv 2742 . . . . . . 7  |-  ( ( S `  m )  e.  _V  ->  ( A. y ( F `  y )  e.  _V  ->  ( F `  ( S `  m )
)  e.  _V )
)
127, 8, 11sylc 62 . . . . . 6  |-  ( ph  ->  ( F `  ( S `  m )
)  e.  _V )
13 ssexg 4025 . . . . . 6  |-  ( ( { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  C_  ( F `  ( S `  m ) )  /\  ( F `  ( S `
 m ) )  e.  _V )  ->  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
143, 12, 13sylancr 408 . . . . 5  |-  ( ph  ->  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  e.  _V )
1514ralrimivw 2478 . . . 4  |-  ( ph  ->  A. m  e.  om  { x  |  ( dom 
S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
16 abrexex2g 5969 . . . 4  |-  ( ( om  e.  _V  /\  A. m  e.  om  {
x  |  ( dom 
S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )  ->  { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
171, 15, 16sylancr 408 . . 3  |-  ( ph  ->  { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
18 simpr 109 . . . . 5  |-  ( ( dom  S  =  (/)  /\  x  e.  A )  ->  x  e.  A
)
1918abssi 3136 . . . 4  |-  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A
20 frecabex.aex . . . 4  |-  ( ph  ->  A  e.  W )
21 ssexg 4025 . . . 4  |-  ( ( { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A  /\  A  e.  W )  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2219, 20, 21sylancr 408 . . 3  |-  ( ph  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2317, 22jca 302 . 2  |-  ( ph  ->  ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  e.  _V  /\ 
{ x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
)
24 unexb 4321 . . 3  |-  ( ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V  /\  { x  |  ( dom 
S  =  (/)  /\  x  e.  A ) }  e.  _V )  <->  ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  u.  {
x  |  ( dom 
S  =  (/)  /\  x  e.  A ) } )  e.  _V )
25 unab 3307 . . . 4  |-  ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m )
) ) }  u.  { x  |  ( dom 
S  =  (/)  /\  x  e.  A ) } )  =  { x  |  ( E. m  e. 
om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m )
) )  \/  ( dom  S  =  (/)  /\  x  e.  A ) ) }
2625eleq1i 2178 . . 3  |-  ( ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  u.  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) } )  e.  _V  <->  { x  |  ( E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m )
) )  \/  ( dom  S  =  (/)  /\  x  e.  A ) ) }  e.  _V )
2724, 26bitri 183 . 2  |-  ( ( { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V  /\  { x  |  ( dom 
S  =  (/)  /\  x  e.  A ) }  e.  _V )  <->  { x  |  ( E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  \/  ( dom 
S  =  (/)  /\  x  e.  A ) ) }  e.  _V )
2823, 27sylib 121 1  |-  ( ph  ->  { x  |  ( E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  \/  ( dom 
S  =  (/)  /\  x  e.  A ) ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680   A.wal 1310    = wceq 1312    e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   _Vcvv 2655    u. cun 3033    C_ wss 3035   (/)c0 3327   suc csuc 4245   omcom 4462   dom cdm 4497   ` cfv 5079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087
This theorem is referenced by:  frectfr  6248
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