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Theorem isf32lem11 8005
Description: Lemma for isfin3-2 8009. Remove hypotheses from isf32lem10 8004. (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 955 . . 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 4473 . . . . . . . 8  |-  ( b  =  c  ->  suc  b  =  suc  c )
32fveq2d 5545 . . . . . . 7  |-  ( b  =  c  ->  ( F `  suc  b )  =  ( F `  suc  c ) )
4 fveq2 5541 . . . . . . 7  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
53, 4sseq12d 3220 . . . . . 6  |-  ( b  =  c  ->  (
( F `  suc  b )  C_  ( F `  b )  <->  ( F `  suc  c
)  C_  ( F `  c ) ) )
65cbvralv 2777 . . . . 5  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  <->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
76biimpi 186 . . . 4  |-  ( A. b  e.  om  ( F `  suc  b ) 
C_  ( F `  b )  ->  A. c  e.  om  ( F `  suc  c )  C_  ( F `  c )
)
873ad2ant2 977 . . 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 957 . . 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 4473 . . . . . 6  |-  ( e  =  d  ->  suc  e  =  suc  d )
1110fveq2d 5545 . . . . 5  |-  ( e  =  d  ->  ( F `  suc  e )  =  ( F `  suc  d ) )
12 fveq2 5541 . . . . 5  |-  ( e  =  d  ->  ( F `  e )  =  ( F `  d ) )
1311, 12psseq12d 3283 . . . 4  |-  ( e  =  d  ->  (
( F `  suc  e )  C.  ( F `  e )  <->  ( F `  suc  d
)  C.  ( F `  d ) ) )
1413cbvrabv 2800 . . 3  |-  { e  e.  om  |  ( F `  suc  e
)  C.  ( F `  e ) }  =  { d  e.  om  |  ( F `  suc  d )  C.  ( F `  d ) }
15 eqid 2296 . . 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 2296 . . 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 2296 . . 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 8004 . 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 419 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165    C. wpss 3166   ~Pcpw 3638   |^|cint 3878   class class class wbr 4039    e. cmpt 4093   suc csuc 4410   omcom 4672   ran crn 4706    o. ccom 4709   iotacio 5233   -->wf 5267   ` cfv 5271   iota_crio 6313    ~~ cen 6876    ~<_* cwdom 7287
This theorem is referenced by:  isf32lem12  8006  fin33i  8011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-wdom 7289  df-card 7588
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