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Theorem isf32lem11 8537
Description: Lemma for isfin3-2 8541. Remove hypotheses from isf32lem10 8536. (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 988 . . 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 4789 . . . . . . . 8  |-  ( b  =  c  ->  suc  b  =  suc  c )
32fveq2d 5700 . . . . . . 7  |-  ( b  =  c  ->  ( F `  suc  b )  =  ( F `  suc  c ) )
4 fveq2 5696 . . . . . . 7  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
53, 4sseq12d 3390 . . . . . 6  |-  ( b  =  c  ->  (
( F `  suc  b )  C_  ( F `  b )  <->  ( F `  suc  c
)  C_  ( F `  c ) ) )
65cbvralv 2952 . . . . 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 1010 . . 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 990 . . 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 4789 . . . . . 6  |-  ( e  =  d  ->  suc  e  =  suc  d )
1110fveq2d 5700 . . . . 5  |-  ( e  =  d  ->  ( F `  suc  e )  =  ( F `  suc  d ) )
12 fveq2 5696 . . . . 5  |-  ( e  =  d  ->  ( F `  e )  =  ( F `  d ) )
1311, 12psseq12d 3455 . . . 4  |-  ( e  =  d  ->  (
( F `  suc  e )  C.  ( F `  e )  <->  ( F `  suc  d
)  C.  ( F `  d ) ) )
1413cbvrabv 2976 . . 3  |-  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  =  { d  e.  om  |  ( F `  suc  d )  C.  ( F `  d ) }
15 eqid 2443 . . 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 2443 . . 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 2443 . . 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 8536 . 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 965    e. wcel 1756   A.wral 2720   {crab 2724    \ cdif 3330    i^i cin 3332    C_ wss 3333    C. wpss 3334   ~Pcpw 3865   |^|cint 4133   class class class wbr 4297    e. cmpt 4355   suc csuc 4726   ran crn 4846    o. ccom 4849   iotacio 5384   -->wf 5419   ` cfv 5423   iota_crio 6056   omcom 6481    ~~ cen 7312    ~<_* cwdom 7777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-om 6482  df-recs 6837  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-wdom 7779  df-card 8114
This theorem is referenced by:  isf32lem12  8538  fin33i  8543
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