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Theorem isf32lem11 8199
Description: Lemma for isfin3-2 8203. Remove hypotheses from isf32lem10 8198. (Contributed by Stefan O'Rear, 17-May-2015.)
Assertion
Ref Expression
isf32lem11  |-  ( ( G  e.  V  /\  ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F ) )  ->  om  ~<_*  G )
Distinct variable groups:    F, b    G, b
Allowed substitution hint:    V( b)

Proof of Theorem isf32lem11
Dummy variables  c 
d  e  f  g  h  k  l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F )  ->  F : om
--> ~P G )
2 suceq 4606 . . . . . . . 8  |-  ( b  =  c  ->  suc  b  =  suc  c )
32fveq2d 5691 . . . . . . 7  |-  ( b  =  c  ->  ( F `  suc  b )  =  ( F `  suc  c ) )
4 fveq2 5687 . . . . . . 7  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
53, 4sseq12d 3337 . . . . . 6  |-  ( b  =  c  ->  (
( F `  suc  b )  C_  ( F `  b )  <->  ( F `  suc  c
)  C_  ( F `  c ) ) )
65cbvralv 2892 . . . . 5  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  <->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
76biimpi 187 . . . 4  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  ->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
873ad2ant2 979 . . 3  |-  ( ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F )  ->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
9 simp3 959 . . 3  |-  ( ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F )  ->  -.  |^| ran  F  e.  ran  F )
10 suceq 4606 . . . . . 6  |-  ( e  =  d  ->  suc  e  =  suc  d )
1110fveq2d 5691 . . . . 5  |-  ( e  =  d  ->  ( F `  suc  e )  =  ( F `  suc  d ) )
12 fveq2 5687 . . . . 5  |-  ( e  =  d  ->  ( F `  e )  =  ( F `  d ) )
1311, 12psseq12d 3401 . . . 4  |-  ( e  =  d  ->  (
( F `  suc  e )  C.  ( F `  e )  <->  ( F `  suc  d
)  C.  ( F `  d ) ) )
1413cbvrabv 2915 . . 3  |-  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  =  { d  e.  om  |  ( F `  suc  d )  C.  ( F `  d ) }
15 eqid 2404 . . 3  |-  ( f  e.  om  |->  ( iota_ g  e.  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) }  ( g  i^i  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) } )  ~~  f ) )  =  ( f  e.  om  |->  ( iota_ g  e.  {
e  e.  om  | 
( F `  suc  e )  C.  ( F `  e ) }  ( g  i^i 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) } )  ~~  f
) )
16 eqid 2404 . . 3  |-  ( ( h  e.  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  |->  ( ( F `  h
)  \  ( F `  suc  h ) ) )  o.  ( f  e.  om  |->  ( iota_ g  e.  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) }  ( g  i^i  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) } )  ~~  f ) ) )  =  ( ( h  e.  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) }  |->  ( ( F `  h ) 
\  ( F `  suc  h ) ) )  o.  ( f  e. 
om  |->  ( iota_ g  e. 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) }  ( g  i^i 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) } )  ~~  f
) ) )
17 eqid 2404 . . 3  |-  ( k  e.  G  |->  ( iota l ( l  e. 
om  /\  k  e.  ( ( ( h  e.  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) }  |->  ( ( F `  h ) 
\  ( F `  suc  h ) ) )  o.  ( f  e. 
om  |->  ( iota_ g  e. 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) }  ( g  i^i 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) } )  ~~  f
) ) ) `  l ) ) ) )  =  ( k  e.  G  |->  ( iota l ( l  e. 
om  /\  k  e.  ( ( ( h  e.  { e  e. 
om  |  ( F `
 suc  e )  C.  ( F `  e
) }  |->  ( ( F `  h ) 
\  ( F `  suc  h ) ) )  o.  ( f  e. 
om  |->  ( iota_ g  e. 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) }  ( g  i^i 
{ e  e.  om  |  ( F `  suc  e )  C.  ( F `  e ) } )  ~~  f
) ) ) `  l ) ) ) )
181, 8, 9, 14, 15, 16, 17isf32lem10 8198 . 2  |-  ( ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F )  ->  ( G  e.  V  ->  om  ~<_*  G ) )
1918impcom 420 1  |-  ( ( G  e.  V  /\  ( F : om --> ~P G  /\  A. b  e.  om  ( F `  suc  b
)  C_  ( F `  b )  /\  -.  |^|
ran  F  e.  ran  F ) )  ->  om  ~<_*  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   A.wral 2666   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280    C. wpss 3281   ~Pcpw 3759   |^|cint 4010   class class class wbr 4172    e. cmpt 4226   suc csuc 4543   omcom 4804   ran crn 4838    o. ccom 4841   iotacio 5375   -->wf 5409   ` cfv 5413   iota_crio 6501    ~~ cen 7065    ~<_* cwdom 7481
This theorem is referenced by:  isf32lem12  8200  fin33i  8205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-wdom 7483  df-card 7782
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