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Theorem isf32lem6 7984
Description: Lemma for isfin3-2 7993. 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 5526 . . 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 3258 . . . . . . . 8  |-  { y  e.  om  |  ( F `  suc  y
)  C.  ( F `  y ) }  C_  om
53, 4eqsstri 3208 . . . . . . 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 7983 . . . . . . 7  |-  ( ph  ->  -.  S  e.  Fin )
10 isf32lem.e . . . . . . . 8  |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S
( v  i^i  S
)  ~~  u )
)
1110fin23lem22 7953 . . . . . . 7  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  J : om -1-1-onto-> S )
125, 9, 11sylancr 644 . . . . . 6  |-  ( ph  ->  J : om -1-1-onto-> S )
13 f1of 5472 . . . . . 6  |-  ( J : om -1-1-onto-> S  ->  J : om
--> S )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  J : om --> S )
15 fvco3 5596 . . . . 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 457 . . . 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 451 . . . . . . . 8  |-  ( (
ph  /\  A  e.  om )  ->  -.  S  e.  Fin )
185, 17, 11sylancr 644 . . . . . . 7  |-  ( (
ph  /\  A  e.  om )  ->  J : om
-1-1-onto-> S )
1918, 13syl 15 . . . . . 6  |-  ( (
ph  /\  A  e.  om )  ->  J : om
--> S )
20 ffvelrn 5663 . . . . . 6  |-  ( ( J : om --> S  /\  A  e.  om )  ->  ( J `  A
)  e.  S )
2119, 20sylancom 648 . . . . 5  |-  ( (
ph  /\  A  e.  om )  ->  ( J `  A )  e.  S
)
22 fveq2 5525 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  w )  =  ( F `  ( J `  A ) ) )
23 suceq 4457 . . . . . . . 8  |-  ( w  =  ( J `  A )  ->  suc  w  =  suc  ( J `
 A ) )
2423fveq2d 5529 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  suc  w )  =  ( F `  suc  ( J `  A
) ) )
2522, 24difeq12d 3295 . . . . . 6  |-  ( w  =  ( J `  A )  ->  (
( F `  w
)  \  ( F `  suc  w ) )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
26 eqid 2283 . . . . . 6  |-  ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) )  =  ( w  e.  S  |->  ( ( F `
 w )  \ 
( F `  suc  w ) ) )
27 fvex 5539 . . . . . . 7  |-  ( F `
 ( J `  A ) )  e. 
_V
28 difexg 4162 . . . . . . 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 5602 . . . . 5  |-  ( ( J `  A )  e.  S  ->  (
( w  e.  S  |->  ( ( F `  w )  \  ( F `  suc  w ) ) ) `  ( J `  A )
)  =  ( ( F `  ( J `
 A ) ) 
\  ( F `  suc  ( J `  A
) ) ) )
3121, 30syl 15 . . . 4  |-  ( (
ph  /\  A  e.  om )  ->  ( (
w  e.  S  |->  ( ( F `  w
)  \  ( F `  suc  w ) ) ) `  ( J `
 A ) )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
3216, 31eqtrd 2315 . . 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 2327 . 2  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =  ( ( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) ) )
34 suceq 4457 . . . . . . . . 9  |-  ( y  =  ( J `  A )  ->  suc  y  =  suc  ( J `
 A ) )
3534fveq2d 5529 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  suc  y )  =  ( F `  suc  ( J `  A
) ) )
36 fveq2 5525 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  y )  =  ( F `  ( J `  A ) ) )
3735, 36psseq12d 3270 . . . . . . 7  |-  ( y  =  ( J `  A )  ->  (
( F `  suc  y )  C.  ( F `  y )  <->  ( F `  suc  ( J `  A )
)  C.  ( F `  ( J `  A
) ) ) )
3837, 3elrab2 2925 . . . . . 6  |-  ( ( J `  A )  e.  S  <->  ( ( J `  A )  e.  om  /\  ( F `
 suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) ) )
3938simprbi 450 . . . . 5  |-  ( ( J `  A )  e.  S  ->  ( F `  suc  ( J `
 A ) ) 
C.  ( F `  ( J `  A ) ) )
4021, 39syl 15 . . . 4  |-  ( (
ph  /\  A  e.  om )  ->  ( F `  suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) )
41 df-pss 3168 . . . 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 188 . . 3  |-  ( (
ph  /\  A  e.  om )  ->  ( ( F `  suc  ( J `
 A ) ) 
C_  ( F `  ( J `  A ) )  /\  ( F `
 suc  ( J `  A ) )  =/=  ( F `  ( J `  A )
) ) )
43 pssdifn0 3515 . . 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 15 . 2  |-  ( (
ph  /\  A  e.  om )  ->  ( ( F `  ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) )  =/=  (/) )
4533, 44eqnetrd 2464 1  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   class class class wbr 4023    e. cmpt 4077   suc csuc 4394   omcom 4656   ran crn 4690    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   iota_crio 6297    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  isf32lem9  7987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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