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Theorem frecabex 6098
Description: The class abstraction from df-frec 6091 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 4374 . . . 4  |-  om  e.  _V
2 simpr 108 . . . . . . 7  |-  ( ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  ->  x  e.  ( F `  ( S `
 m ) ) )
32abssi 3082 . . . . . 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 2617 . . . . . . . 8  |-  m  e. 
_V
6 fvexg 5272 . . . . . . . 8  |-  ( ( S  e.  V  /\  m  e.  _V )  ->  ( S `  m
)  e.  _V )
74, 5, 6sylancl 404 . . . . . . 7  |-  ( ph  ->  ( S `  m
)  e.  _V )
8 frecabex.fvex . . . . . . 7  |-  ( ph  ->  A. y ( F `
 y )  e. 
_V )
9 fveq2 5256 . . . . . . . . 9  |-  ( y  =  ( S `  m )  ->  ( F `  y )  =  ( F `  ( S `  m ) ) )
109eleq1d 2153 . . . . . . . 8  |-  ( y  =  ( S `  m )  ->  (
( F `  y
)  e.  _V  <->  ( F `  ( S `  m
) )  e.  _V ) )
1110spcgv 2698 . . . . . . 7  |-  ( ( S `  m )  e.  _V  ->  ( A. y ( F `  y )  e.  _V  ->  ( F `  ( S `  m )
)  e.  _V )
)
127, 8, 11sylc 61 . . . . . 6  |-  ( ph  ->  ( F `  ( S `  m )
)  e.  _V )
13 ssexg 3946 . . . . . 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 405 . . . . 5  |-  ( ph  ->  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  e.  _V )
1514ralrimivw 2443 . . . 4  |-  ( ph  ->  A. m  e.  om  { x  |  ( dom 
S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
16 abrexex2g 5829 . . . 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 405 . . 3  |-  ( ph  ->  { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
18 simpr 108 . . . . 5  |-  ( ( dom  S  =  (/)  /\  x  e.  A )  ->  x  e.  A
)
1918abssi 3082 . . . 4  |-  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A
20 frecabex.aex . . . 4  |-  ( ph  ->  A  e.  W )
21 ssexg 3946 . . . 4  |-  ( ( { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A  /\  A  e.  W )  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2219, 20, 21sylancr 405 . . 3  |-  ( ph  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2317, 22jca 300 . 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 4234 . . 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 3252 . . . 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 2150 . . 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 182 . 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 120 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 102    \/ wo 662   A.wal 1285    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2614    u. cun 2984    C_ wss 2986   (/)c0 3272   suc csuc 4159   omcom 4371   dom cdm 4404   ` cfv 4972
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-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-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  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-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980
This theorem is referenced by:  frectfr  6100
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