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Theorem frecabex 6117
Description: The class abstraction from df-frec 6110 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 4381 . . . 4  |-  om  e.  _V
2 simpr 108 . . . . . . 7  |-  ( ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  ->  x  e.  ( F `  ( S `
 m ) ) )
32abssi 3085 . . . . . 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 2618 . . . . . . . 8  |-  m  e. 
_V
6 fvexg 5287 . . . . . . . 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 5268 . . . . . . . . 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 2699 . . . . . . 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 3953 . . . . . 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 5848 . . . 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 3085 . . . 4  |-  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A
20 frecabex.aex . . . 4  |-  ( ph  ->  A  e.  W )
21 ssexg 3953 . . . 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 4241 . . 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 3255 . . . 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 2615    u. cun 2986    C_ wss 2988   (/)c0 3275   suc csuc 4166   omcom 4378   dom cdm 4411   ` cfv 4981
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 3929  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-iinf 4376
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 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989
This theorem is referenced by:  frectfr  6119
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