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Theorem isf32lem6 8238
Description: Lemma for isfin3-2 8247. Each K value is non-empty. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a  |-  ( ph  ->  F : om --> ~P G
)
isf32lem.b  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
isf32lem.c  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
isf32lem.d  |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `  y ) }
isf32lem.e  |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S
( v  i^i  S
)  ~~  u )
)
isf32lem.f  |-  K  =  ( ( w  e.  S  |->  ( ( F `
 w )  \ 
( F `  suc  w ) ) )  o.  J )
Assertion
Ref Expression
isf32lem6  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
Distinct variable groups:    x, w    v, u, w, x, y,
ph    w, A, x, y   
w, F, x, y   
u, S, v, w, x, y    w, J, x, y    x, K, y
Allowed substitution hints:    A( v, u)    F( v, u)    G( x, y, w, v, u)    J( v, u)    K( w, v, u)

Proof of Theorem isf32lem6
StepHypRef Expression
1 isf32lem.f . . . 4  |-  K  =  ( ( w  e.  S  |->  ( ( F `
 w )  \ 
( F `  suc  w ) ) )  o.  J )
21fveq1i 5729 . . 3  |-  ( K `
 A )  =  ( ( ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) )  o.  J ) `  A )
3 isf32lem.d . . . . . . . 8  |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `  y ) }
4 ssrab2 3428 . . . . . . . 8  |-  { y  e.  om  |  ( F `  suc  y
)  C.  ( F `  y ) }  C_  om
53, 4eqsstri 3378 . . . . . . 7  |-  S  C_  om
6 isf32lem.a . . . . . . . 8  |-  ( ph  ->  F : om --> ~P G
)
7 isf32lem.b . . . . . . . 8  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
8 isf32lem.c . . . . . . . 8  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
96, 7, 8, 3isf32lem5 8237 . . . . . . 7  |-  ( ph  ->  -.  S  e.  Fin )
10 isf32lem.e . . . . . . . 8  |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S
( v  i^i  S
)  ~~  u )
)
1110fin23lem22 8207 . . . . . . 7  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  J : om -1-1-onto-> S )
125, 9, 11sylancr 645 . . . . . 6  |-  ( ph  ->  J : om -1-1-onto-> S )
13 f1of 5674 . . . . . 6  |-  ( J : om -1-1-onto-> S  ->  J : om
--> S )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  J : om --> S )
15 fvco3 5800 . . . . 5  |-  ( ( J : om --> S  /\  A  e.  om )  ->  ( ( ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) )  o.  J ) `  A )  =  ( ( w  e.  S  |->  ( ( F `  w )  \  ( F `  suc  w ) ) ) `  ( J `  A )
) )
1614, 15sylan 458 . . . 4  |-  ( (
ph  /\  A  e.  om )  ->  ( (
( w  e.  S  |->  ( ( F `  w )  \  ( F `  suc  w ) ) )  o.  J
) `  A )  =  ( ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) ) `
 ( J `  A ) ) )
179adantr 452 . . . . . . . 8  |-  ( (
ph  /\  A  e.  om )  ->  -.  S  e.  Fin )
185, 17, 11sylancr 645 . . . . . . 7  |-  ( (
ph  /\  A  e.  om )  ->  J : om
-1-1-onto-> S )
1918, 13syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  om )  ->  J : om
--> S )
20 ffvelrn 5868 . . . . . 6  |-  ( ( J : om --> S  /\  A  e.  om )  ->  ( J `  A
)  e.  S )
2119, 20sylancom 649 . . . . 5  |-  ( (
ph  /\  A  e.  om )  ->  ( J `  A )  e.  S
)
22 fveq2 5728 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  w )  =  ( F `  ( J `  A ) ) )
23 suceq 4646 . . . . . . . 8  |-  ( w  =  ( J `  A )  ->  suc  w  =  suc  ( J `
 A ) )
2423fveq2d 5732 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  suc  w )  =  ( F `  suc  ( J `  A
) ) )
2522, 24difeq12d 3466 . . . . . 6  |-  ( w  =  ( J `  A )  ->  (
( F `  w
)  \  ( F `  suc  w ) )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
26 eqid 2436 . . . . . 6  |-  ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) )  =  ( w  e.  S  |->  ( ( F `
 w )  \ 
( F `  suc  w ) ) )
27 fvex 5742 . . . . . . 7  |-  ( F `
 ( J `  A ) )  e. 
_V
28 difexg 4351 . . . . . . 7  |-  ( ( F `  ( J `
 A ) )  e.  _V  ->  (
( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) )  e.  _V )
2927, 28ax-mp 8 . . . . . 6  |-  ( ( F `  ( J `
 A ) ) 
\  ( F `  suc  ( J `  A
) ) )  e. 
_V
3025, 26, 29fvmpt 5806 . . . . 5  |-  ( ( J `  A )  e.  S  ->  (
( w  e.  S  |->  ( ( F `  w )  \  ( F `  suc  w ) ) ) `  ( J `  A )
)  =  ( ( F `  ( J `
 A ) ) 
\  ( F `  suc  ( J `  A
) ) ) )
3121, 30syl 16 . . . 4  |-  ( (
ph  /\  A  e.  om )  ->  ( (
w  e.  S  |->  ( ( F `  w
)  \  ( F `  suc  w ) ) ) `  ( J `
 A ) )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
3216, 31eqtrd 2468 . . 3  |-  ( (
ph  /\  A  e.  om )  ->  ( (
( w  e.  S  |->  ( ( F `  w )  \  ( F `  suc  w ) ) )  o.  J
) `  A )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
332, 32syl5eq 2480 . 2  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =  ( ( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) ) )
34 suceq 4646 . . . . . . . . 9  |-  ( y  =  ( J `  A )  ->  suc  y  =  suc  ( J `
 A ) )
3534fveq2d 5732 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  suc  y )  =  ( F `  suc  ( J `  A
) ) )
36 fveq2 5728 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  y )  =  ( F `  ( J `  A ) ) )
3735, 36psseq12d 3441 . . . . . . 7  |-  ( y  =  ( J `  A )  ->  (
( F `  suc  y )  C.  ( F `  y )  <->  ( F `  suc  ( J `  A )
)  C.  ( F `  ( J `  A
) ) ) )
3837, 3elrab2 3094 . . . . . 6  |-  ( ( J `  A )  e.  S  <->  ( ( J `  A )  e.  om  /\  ( F `
 suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) ) )
3938simprbi 451 . . . . 5  |-  ( ( J `  A )  e.  S  ->  ( F `  suc  ( J `
 A ) ) 
C.  ( F `  ( J `  A ) ) )
4021, 39syl 16 . . . 4  |-  ( (
ph  /\  A  e.  om )  ->  ( F `  suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) )
41 df-pss 3336 . . . 4  |-  ( ( F `  suc  ( J `  A )
)  C.  ( F `  ( J `  A
) )  <->  ( ( F `  suc  ( J `
 A ) ) 
C_  ( F `  ( J `  A ) )  /\  ( F `
 suc  ( J `  A ) )  =/=  ( F `  ( J `  A )
) ) )
4240, 41sylib 189 . . 3  |-  ( (
ph  /\  A  e.  om )  ->  ( ( F `  suc  ( J `
 A ) ) 
C_  ( F `  ( J `  A ) )  /\  ( F `
 suc  ( J `  A ) )  =/=  ( F `  ( J `  A )
) ) )
43 pssdifn0 3689 . . 3  |-  ( ( ( F `  suc  ( J `  A ) )  C_  ( F `  ( J `  A
) )  /\  ( F `  suc  ( J `
 A ) )  =/=  ( F `  ( J `  A ) ) )  ->  (
( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) )  =/=  (/) )
4442, 43syl 16 . 2  |-  ( (
ph  /\  A  e.  om )  ->  ( ( F `  ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) )  =/=  (/) )
4533, 44eqnetrd 2619 1  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320    C. wpss 3321   (/)c0 3628   ~Pcpw 3799   |^|cint 4050   class class class wbr 4212    e. cmpt 4266   suc csuc 4583   omcom 4845   ran crn 4879    o. ccom 4882   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454   iota_crio 6542    ~~ cen 7106   Fincfn 7109
This theorem is referenced by:  isf32lem9  8241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826
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