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Theorem isf32lem11 8760
Description: Lemma for isfin3-2 8764. Remove hypotheses from isf32lem10 8759. (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 4952 . . . . . . . 8  |-  ( b  =  c  ->  suc  b  =  suc  c )
32fveq2d 5876 . . . . . . 7  |-  ( b  =  c  ->  ( F `  suc  b )  =  ( F `  suc  c ) )
4 fveq2 5872 . . . . . . 7  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
53, 4sseq12d 3528 . . . . . 6  |-  ( b  =  c  ->  (
( F `  suc  b )  C_  ( F `  b )  <->  ( F `  suc  c
)  C_  ( F `  c ) ) )
65cbvralv 3084 . . . . 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 4952 . . . . . 6  |-  ( e  =  d  ->  suc  e  =  suc  d )
1110fveq2d 5876 . . . . 5  |-  ( e  =  d  ->  ( F `  suc  e )  =  ( F `  suc  d ) )
12 fveq2 5872 . . . . 5  |-  ( e  =  d  ->  ( F `  e )  =  ( F `  d ) )
1311, 12psseq12d 3594 . . . 4  |-  ( e  =  d  ->  (
( F `  suc  e )  C.  ( F `  e )  <->  ( F `  suc  d
)  C.  ( F `  d ) ) )
1413cbvrabv 3108 . . 3  |-  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  =  { d  e.  om  |  ( F `  suc  d )  C.  ( F `  d ) }
15 eqid 2457 . . 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 2457 . . 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 2457 . . 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 8759 . 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 1819   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471    C. wpss 3472   ~Pcpw 4015   |^|cint 4288   class class class wbr 4456    |-> cmpt 4515   suc csuc 4889   ran crn 5009    o. ccom 5012   iotacio 5555   -->wf 5590   ` cfv 5594   iota_crio 6257   omcom 6699    ~~ cen 7532    ~<_* cwdom 8001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-wdom 8003  df-card 8337
This theorem is referenced by:  isf32lem12  8761  fin33i  8766
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