MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf32lem6 Structured version   Unicode version

Theorem isf32lem6 8639
Description: Lemma for isfin3-2 8648. 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 5801 . . 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 3546 . . . . . . . 8  |-  { y  e.  om  |  ( F `  suc  y
)  C.  ( F `  y ) }  C_  om
53, 4eqsstri 3495 . . . . . . 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 8638 . . . . . . 7  |-  ( ph  ->  -.  S  e.  Fin )
10 isf32lem.e . . . . . . . 8  |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S  ( v  i^i  S
)  ~~  u )
)
1110fin23lem22 8608 . . . . . . 7  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  J : om -1-1-onto-> S )
125, 9, 11sylancr 663 . . . . . 6  |-  ( ph  ->  J : om -1-1-onto-> S )
13 f1of 5750 . . . . . 6  |-  ( J : om -1-1-onto-> S  ->  J : om
--> S )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  J : om --> S )
15 fvco3 5878 . . . . 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 471 . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  A  e.  om )  ->  -.  S  e.  Fin )
185, 17, 11sylancr 663 . . . . . . 7  |-  ( (
ph  /\  A  e.  om )  ->  J : om
-1-1-onto-> S )
1918, 13syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  om )  ->  J : om
--> S )
20 ffvelrn 5951 . . . . . 6  |-  ( ( J : om --> S  /\  A  e.  om )  ->  ( J `  A
)  e.  S )
2119, 20sylancom 667 . . . . 5  |-  ( (
ph  /\  A  e.  om )  ->  ( J `  A )  e.  S
)
22 fveq2 5800 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  w )  =  ( F `  ( J `  A ) ) )
23 suceq 4893 . . . . . . . 8  |-  ( w  =  ( J `  A )  ->  suc  w  =  suc  ( J `
 A ) )
2423fveq2d 5804 . . . . . . 7  |-  ( w  =  ( J `  A )  ->  ( F `  suc  w )  =  ( F `  suc  ( J `  A
) ) )
2522, 24difeq12d 3584 . . . . . 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 5810 . . . . . . 7  |-  ( F `
 ( J `  A ) )  e. 
_V
28 difexg 4549 . . . . . . 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 5884 . . . . 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 4893 . . . . . . . . 9  |-  ( y  =  ( J `  A )  ->  suc  y  =  suc  ( J `
 A ) )
3534fveq2d 5804 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  suc  y )  =  ( F `  suc  ( J `  A
) ) )
36 fveq2 5800 . . . . . . . 8  |-  ( y  =  ( J `  A )  ->  ( F `  y )  =  ( F `  ( J `  A ) ) )
3735, 36psseq12d 3559 . . . . . . 7  |-  ( y  =  ( J `  A )  ->  (
( F `  suc  y )  C.  ( F `  y )  <->  ( F `  suc  ( J `  A )
)  C.  ( F `  ( J `  A
) ) ) )
3837, 3elrab2 3226 . . . . . 6  |-  ( ( J `  A )  e.  S  <->  ( ( J `  A )  e.  om  /\  ( F `
 suc  ( J `  A ) )  C.  ( F `  ( J `
 A ) ) ) )
3938simprbi 464 . . . . 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 3453 . . . 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 3849 . . 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 2745 1  |-  ( (
ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078    \ cdif 3434    i^i cin 3436    C_ wss 3437    C. wpss 3438   (/)c0 3746   ~Pcpw 3969   |^|cint 4237   class class class wbr 4401    |-> cmpt 4459   suc csuc 4830   ran crn 4950    o. ccom 4953   -->wf 5523   -1-1-onto->wf1o 5526   ` cfv 5527   iota_crio 6161   omcom 6587    ~~ cen 7418   Fincfn 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-om 6588  df-recs 6943  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221
This theorem is referenced by:  isf32lem9  8642
  Copyright terms: Public domain W3C validator