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Theorem isf32lem6 8729
Description: Lemma for isfin3-2 8738. Each K value is nonempty. (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 5849 . . 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 3571 . . . . . . . 8  |-  { y  e.  om  |  ( F `  suc  y
)  C.  ( F `  y ) }  C_  om
53, 4eqsstri 3519 . . . . . . 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 8728 . . . . . . 7  |-  ( ph  ->  -.  S  e.  Fin )
10 isf32lem.e . . . . . . . 8  |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S  ( v  i^i  S
)  ~~  u )
)
1110fin23lem22 8698 . . . . . . 7  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  J : om -1-1-onto-> S )
125, 9, 11sylancr 661 . . . . . 6  |-  ( ph  ->  J : om -1-1-onto-> S )
13 f1of 5798 . . . . . 6  |-  ( J : om -1-1-onto-> S  ->  J : om
--> S )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  J : om --> S )
15 fvco3 5925 . . . . 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 469 . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  A  e.  om )  ->  -.  S  e.  Fin )
185, 17, 11sylancr 661 . . . . . . 7  |-  ( (
ph  /\  A  e.  om )  ->  J : om
-1-1-onto-> S )
1918, 13syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  om )  ->  J : om
--> S )
20 ffvelrn 6005 . . . . . 6  |-  ( ( J : om --> S  /\  A  e.  om )  ->  ( J `  A
)  e.  S )
2119, 20sylancom 665 . . . . 5  |-  ( (
ph  /\  A  e.  om )  ->  ( J `  A )  e.  S
)
22 fveq2 5848 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  w )  =  ( F `  ( J `  A ) ) )
23 suceq 4932 . . . . . . . 8  |-  ( w  =  ( J `  A )  ->  suc  w  =  suc  ( J `
 A ) )
2423fveq2d 5852 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  suc  w )  =  ( F `  suc  ( J `  A
) ) )
2522, 24difeq12d 3609 . . . . . 6  |-  ( w  =  ( J `  A )  ->  (
( F `  w
)  \  ( F `  suc  w ) )  =  ( ( F `
 ( J `  A ) )  \ 
( F `  suc  ( J `  A ) ) ) )
26 eqid 2454 . . . . . 6  |-  ( w  e.  S  |->  ( ( F `  w ) 
\  ( F `  suc  w ) ) )  =  ( w  e.  S  |->  ( ( F `
 w )  \ 
( F `  suc  w ) ) )
27 fvex 5858 . . . . . . 7  |-  ( F `
 ( J `  A ) )  e. 
_V
28 difexg 4585 . . . . . . 7  |-  ( ( F `  ( J `
 A ) )  e.  _V  ->  (
( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) )  e.  _V )
2927, 28ax-mp 5 . . . . . 6  |-  ( ( F `  ( J `
 A ) ) 
\  ( F `  suc  ( J `  A
) ) )  e. 
_V
3025, 26, 29fvmpt 5931 . . . . 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 2495 . . 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 2507 . 2  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =  ( ( F `  ( J `  A )
)  \  ( F `  suc  ( J `  A ) ) ) )
34 suceq 4932 . . . . . . . . 9  |-  ( y  =  ( J `  A )  ->  suc  y  =  suc  ( J `
 A ) )
3534fveq2d 5852 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  suc  y )  =  ( F `  suc  ( J `  A
) ) )
36 fveq2 5848 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  y )  =  ( F `  ( J `  A ) ) )
3735, 36psseq12d 3584 . . . . . . 7  |-  ( y  =  ( J `  A )  ->  (
( F `  suc  y )  C.  ( F `  y )  <->  ( F `  suc  ( J `  A )
)  C.  ( F `  ( J `  A
) ) ) )
3837, 3elrab2 3256 . . . . . 6  |-  ( ( J `  A )  e.  S  <->  ( ( J `  A )  e.  om  /\  ( F `
 suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) ) )
3938simprbi 462 . . . . 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 3477 . . . 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 196 . . 3  |-  ( (
ph  /\  A  e.  om )  ->  ( ( F `  suc  ( J `
 A ) ) 
C_  ( F `  ( J `  A ) )  /\  ( F `
 suc  ( J `  A ) )  =/=  ( F `  ( J `  A )
) ) )
43 pssdifn0 3876 . . 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 2747 1  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3783   ~Pcpw 3999   |^|cint 4271   class class class wbr 4439    |-> cmpt 4497   suc csuc 4869   ran crn 4989    o. ccom 4992   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570   iota_crio 6231   omcom 6673    ~~ cen 7506   Fincfn 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311
This theorem is referenced by:  isf32lem9  8732
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