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Theorem frecabex 6392
Description: The class abstraction from df-frec 6385 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 4588 . . . 4  |-  om  e.  _V
2 simpr 110 . . . . . . 7  |-  ( ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  ->  x  e.  ( F `  ( S `
 m ) ) )
32abssi 3230 . . . . . 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 2740 . . . . . . . 8  |-  m  e. 
_V
6 fvexg 5529 . . . . . . . 8  |-  ( ( S  e.  V  /\  m  e.  _V )  ->  ( S `  m
)  e.  _V )
74, 5, 6sylancl 413 . . . . . . 7  |-  ( ph  ->  ( S `  m
)  e.  _V )
8 frecabex.fvex . . . . . . 7  |-  ( ph  ->  A. y ( F `
 y )  e. 
_V )
9 fveq2 5510 . . . . . . . . 9  |-  ( y  =  ( S `  m )  ->  ( F `  y )  =  ( F `  ( S `  m ) ) )
109eleq1d 2246 . . . . . . . 8  |-  ( y  =  ( S `  m )  ->  (
( F `  y
)  e.  _V  <->  ( F `  ( S `  m
) )  e.  _V ) )
1110spcgv 2824 . . . . . . 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 4139 . . . . . 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 414 . . . . 5  |-  ( ph  ->  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  e.  _V )
1514ralrimivw 2551 . . . 4  |-  ( ph  ->  A. m  e.  om  { x  |  ( dom 
S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
16 abrexex2g 6114 . . . 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 414 . . 3  |-  ( ph  ->  { x  |  E. m  e.  om  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
18 simpr 110 . . . . 5  |-  ( ( dom  S  =  (/)  /\  x  e.  A )  ->  x  e.  A
)
1918abssi 3230 . . . 4  |-  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A
20 frecabex.aex . . . 4  |-  ( ph  ->  A  e.  W )
21 ssexg 4139 . . . 4  |-  ( ( { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A  /\  A  e.  W )  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2219, 20, 21sylancr 414 . . 3  |-  ( ph  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2317, 22jca 306 . 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 4438 . . 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 3402 . . . 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 2243 . . 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 184 . 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 122 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 104    \/ wo 708   A.wal 1351    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737    u. cun 3127    C_ wss 3129   (/)c0 3422   suc csuc 4361   omcom 4585   dom cdm 4622   ` cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219
This theorem is referenced by:  frectfr  6394
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