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Theorem frecabex 6302
Description: The class abstraction from df-frec 6295 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 4514 . . . 4  |-  om  e.  _V
2 simpr 109 . . . . . . 7  |-  ( ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) )  ->  x  e.  ( F `  ( S `
 m ) ) )
32abssi 3176 . . . . . 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 2692 . . . . . . . 8  |-  m  e. 
_V
6 fvexg 5447 . . . . . . . 8  |-  ( ( S  e.  V  /\  m  e.  _V )  ->  ( S `  m
)  e.  _V )
74, 5, 6sylancl 410 . . . . . . 7  |-  ( ph  ->  ( S `  m
)  e.  _V )
8 frecabex.fvex . . . . . . 7  |-  ( ph  ->  A. y ( F `
 y )  e. 
_V )
9 fveq2 5428 . . . . . . . . 9  |-  ( y  =  ( S `  m )  ->  ( F `  y )  =  ( F `  ( S `  m ) ) )
109eleq1d 2209 . . . . . . . 8  |-  ( y  =  ( S `  m )  ->  (
( F `  y
)  e.  _V  <->  ( F `  ( S `  m
) )  e.  _V ) )
1110spcgv 2776 . . . . . . 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 4074 . . . . . 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 411 . . . . 5  |-  ( ph  ->  { x  |  ( dom  S  =  suc  m  /\  x  e.  ( F `  ( S `
 m ) ) ) }  e.  _V )
1514ralrimivw 2509 . . . 4  |-  ( ph  ->  A. m  e.  om  { x  |  ( dom 
S  =  suc  m  /\  x  e.  ( F `  ( S `  m ) ) ) }  e.  _V )
16 abrexex2g 6025 . . . 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 411 . . 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 3176 . . . 4  |-  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A
20 frecabex.aex . . . 4  |-  ( ph  ->  A  e.  W )
21 ssexg 4074 . . . 4  |-  ( ( { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  C_  A  /\  A  e.  W )  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2219, 20, 21sylancr 411 . . 3  |-  ( ph  ->  { x  |  ( dom  S  =  (/)  /\  x  e.  A ) }  e.  _V )
2317, 22jca 304 . 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 4370 . . 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 3347 . . . 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 2206 . . 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 698   A.wal 1330    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   _Vcvv 2689    u. cun 3073    C_ wss 3075   (/)c0 3367   suc csuc 4294   omcom 4511   dom cdm 4546   ` cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138
This theorem is referenced by:  frectfr  6304
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