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Theorem isf32lem11 8739
Description: Lemma for isfin3-2 8743. Remove hypotheses from isf32lem10 8738. (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 996 . . 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 4943 . . . . . . . 8  |-  ( b  =  c  ->  suc  b  =  suc  c )
32fveq2d 5868 . . . . . . 7  |-  ( b  =  c  ->  ( F `  suc  b )  =  ( F `  suc  c ) )
4 fveq2 5864 . . . . . . 7  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
53, 4sseq12d 3533 . . . . . 6  |-  ( b  =  c  ->  (
( F `  suc  b )  C_  ( F `  b )  <->  ( F `  suc  c
)  C_  ( F `  c ) ) )
65cbvralv 3088 . . . . 5  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  <->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
76biimpi 194 . . . 4  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  ->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
873ad2ant2 1018 . . 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 998 . . 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 4943 . . . . . 6  |-  ( e  =  d  ->  suc  e  =  suc  d )
1110fveq2d 5868 . . . . 5  |-  ( e  =  d  ->  ( F `  suc  e )  =  ( F `  suc  d ) )
12 fveq2 5864 . . . . 5  |-  ( e  =  d  ->  ( F `  e )  =  ( F `  d ) )
1311, 12psseq12d 3598 . . . 4  |-  ( e  =  d  ->  (
( F `  suc  e )  C.  ( F `  e )  <->  ( F `  suc  d
)  C.  ( F `  d ) ) )
1413cbvrabv 3112 . . 3  |-  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  =  { d  e.  om  |  ( F `  suc  d )  C.  ( F `  d ) }
15 eqid 2467 . . 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 2467 . . 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 2467 . . 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 8738 . 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 430 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476    C. wpss 3477   ~Pcpw 4010   |^|cint 4282   class class class wbr 4447    |-> cmpt 4505   suc csuc 4880   ran crn 5000    o. ccom 5003   iotacio 5547   -->wf 5582   ` cfv 5586   iota_crio 6242   omcom 6678    ~~ cen 7510    ~<_* cwdom 7979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-om 6679  df-recs 7039  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316
This theorem is referenced by:  isf32lem12  8740  fin33i  8745
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