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Theorem filnetlem4 31087
Description: Lemma for filnet 31088. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h  |-  H  = 
U_ n  e.  F  ( { n }  X.  n )
filnet.d  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
Assertion
Ref Expression
filnetlem4  |-  ( F  e.  ( Fil `  X
)  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `
 ran  ( tail `  d ) ) ) )
Distinct variable groups:    x, y    f, d, n, x, y, F    H, d, f, x, y    D, d, f    X, d, f, n
Allowed substitution hints:    D( x, y, n)    H( n)    X( x, y)

Proof of Theorem filnetlem4
Dummy variables  k  m  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filnet.h . . . . 5  |-  H  = 
U_ n  e.  F  ( { n }  X.  n )
2 filnet.d . . . . 5  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
31, 2filnetlem3 31086 . . . 4  |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) ) )
43simpri 468 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) )
54simprd 469 . 2  |-  ( F  e.  ( Fil `  X
)  ->  D  e.  DirRel )
6 f2ndres 6848 . . . . 5  |-  ( 2nd  |`  ( F  X.  X
) ) : ( F  X.  X ) --> X
74simpld 465 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  H  C_  ( F  X.  X ) )
8 fssres2 5778 . . . . 5  |-  ( ( ( 2nd  |`  ( F  X.  X ) ) : ( F  X.  X ) --> X  /\  H  C_  ( F  X.  X ) )  -> 
( 2nd  |`  H ) : H --> X )
96, 7, 8sylancr 674 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( 2nd  |`  H ) : H --> X )
10 filtop 20925 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
11 xpexg 6625 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  X  e.  F )  ->  ( F  X.  X )  e. 
_V )
1210, 11mpdan 679 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( F  X.  X )  e.  _V )
1312, 7ssexd 4566 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  H  e.  _V )
14 fex 6168 . . . 4  |-  ( ( ( 2nd  |`  H ) : H --> X  /\  H  e.  _V )  ->  ( 2nd  |`  H )  e.  _V )
159, 13, 14syl2anc 671 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( 2nd  |`  H )  e.  _V )
16 dirdm 16535 . . . . . . . 8  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
175, 16syl 17 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  dom  D  = 
U. U. D )
183simpli 464 . . . . . . 7  |-  H  = 
U. U. D
1917, 18syl6reqr 2515 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  H  =  dom  D )
2019feq2d 5741 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( ( 2nd  |`  H ) : H --> X  <->  ( 2nd  |`  H ) : dom  D --> X ) )
219, 20mpbid 215 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( 2nd  |`  H ) : dom  D --> X )
22 eqid 2462 . . . . . . . . . . . . . 14  |-  dom  D  =  dom  D
2322tailf 31081 . . . . . . . . . . . . 13  |-  ( D  e.  DirRel  ->  ( tail `  D
) : dom  D --> ~P dom  D )
245, 23syl 17 . . . . . . . . . . . 12  |-  ( F  e.  ( Fil `  X
)  ->  ( tail `  D ) : dom  D --> ~P dom  D )
2519feq2d 5741 . . . . . . . . . . . 12  |-  ( F  e.  ( Fil `  X
)  ->  ( ( tail `  D ) : H --> ~P dom  D  <->  (
tail `  D ) : dom  D --> ~P dom  D ) )
2624, 25mpbird 240 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  ( tail `  D ) : H --> ~P dom  D )
2726adantr 471 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( tail `  D ) : H --> ~P dom  D
)
28 ffn 5755 . . . . . . . . . 10  |-  ( (
tail `  D ) : H --> ~P dom  D  ->  ( tail `  D
)  Fn  H )
29 imaeq2 5186 . . . . . . . . . . . 12  |-  ( d  =  ( ( tail `  D ) `  f
)  ->  ( ( 2nd  |`  H ) "
d )  =  ( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) ) )
3029sseq1d 3471 . . . . . . . . . . 11  |-  ( d  =  ( ( tail `  D ) `  f
)  ->  ( (
( 2nd  |`  H )
" d )  C_  t 
<->  ( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t
) )
3130rexrn 6052 . . . . . . . . . 10  |-  ( (
tail `  D )  Fn  H  ->  ( E. d  e.  ran  ( tail `  D ) ( ( 2nd  |`  H )
" d )  C_  t 
<->  E. f  e.  H  ( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t
) )
3227, 28, 313syl 18 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( E. d  e.  ran  ( tail `  D )
( ( 2nd  |`  H )
" d )  C_  t 
<->  E. f  e.  H  ( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t
) )
33 fo2nd 6846 . . . . . . . . . . . . . . 15  |-  2nd : _V -onto-> _V
34 fofn 5822 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
3533, 34ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd  Fn  _V
36 ssv 3464 . . . . . . . . . . . . . 14  |-  H  C_  _V
37 fnssres 5715 . . . . . . . . . . . . . 14  |-  ( ( 2nd  Fn  _V  /\  H  C_  _V )  -> 
( 2nd  |`  H )  Fn  H )
3835, 36, 37mp2an 683 . . . . . . . . . . . . 13  |-  ( 2nd  |`  H )  Fn  H
39 fnfun 5699 . . . . . . . . . . . . 13  |-  ( ( 2nd  |`  H )  Fn  H  ->  Fun  ( 2nd  |`  H ) )
4038, 39ax-mp 5 . . . . . . . . . . . 12  |-  Fun  ( 2nd  |`  H )
4127ffvelrnda 6050 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( tail `  D ) `  f )  e.  ~P dom  D )
4241elpwid 3973 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( tail `  D ) `  f )  C_  dom  D )
4319ad2antrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  H  =  dom  D )
4442, 43sseqtr4d 3481 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( tail `  D ) `  f )  C_  H
)
45 fndm 5701 . . . . . . . . . . . . . 14  |-  ( ( 2nd  |`  H )  Fn  H  ->  dom  ( 2nd  |`  H )  =  H )
4638, 45ax-mp 5 . . . . . . . . . . . . 13  |-  dom  ( 2nd  |`  H )  =  H
4744, 46syl6sseqr 3491 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( tail `  D ) `  f )  C_  dom  ( 2nd  |`  H )
)
48 funimass4 5943 . . . . . . . . . . . 12  |-  ( ( Fun  ( 2nd  |`  H )  /\  ( ( tail `  D ) `  f
)  C_  dom  ( 2nd  |`  H ) )  -> 
( ( ( 2nd  |`  H ) " (
( tail `  D ) `  f ) )  C_  t 
<-> 
A. d  e.  ( ( tail `  D
) `  f )
( ( 2nd  |`  H ) `
 d )  e.  t ) )
4940, 47, 48sylancr 674 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t  <->  A. d  e.  ( (
tail `  D ) `  f ) ( ( 2nd  |`  H ) `  d )  e.  t ) )
505ad2antrr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  D  e.  DirRel )
51 simpr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  f  e.  H )
5251, 43eleqtrd 2542 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  f  e.  dom  D )
53 vex 3060 . . . . . . . . . . . . . . . . 17  |-  d  e. 
_V
5453a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  d  e.  _V )
5522eltail 31080 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  f  e.  dom  D  /\  d  e.  _V )  ->  (
d  e.  ( (
tail `  D ) `  f )  <->  f D
d ) )
5650, 52, 54, 55syl3anc 1276 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
d  e.  ( (
tail `  D ) `  f )  <->  f D
d ) )
5751biantrurd 515 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
d  e.  H  <->  ( f  e.  H  /\  d  e.  H ) ) )
5857anbi1d 716 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( d  e.  H  /\  ( 1st `  d
)  C_  ( 1st `  f ) )  <->  ( (
f  e.  H  /\  d  e.  H )  /\  ( 1st `  d
)  C_  ( 1st `  f ) ) ) )
59 vex 3060 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
601, 2, 59, 53filnetlem1 31084 . . . . . . . . . . . . . . . 16  |-  ( f D d  <->  ( (
f  e.  H  /\  d  e.  H )  /\  ( 1st `  d
)  C_  ( 1st `  f ) ) )
6158, 60syl6bbr 271 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( d  e.  H  /\  ( 1st `  d
)  C_  ( 1st `  f ) )  <->  f D
d ) )
6256, 61bitr4d 264 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
d  e.  ( (
tail `  D ) `  f )  <->  ( d  e.  H  /\  ( 1st `  d )  C_  ( 1st `  f ) ) ) )
6362imbi1d 323 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( d  e.  ( ( tail `  D
) `  f )  ->  ( ( 2nd  |`  H ) `
 d )  e.  t )  <->  ( (
d  e.  H  /\  ( 1st `  d ) 
C_  ( 1st `  f
) )  ->  (
( 2nd  |`  H ) `
 d )  e.  t ) ) )
64 fvres 5906 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  H  ->  (
( 2nd  |`  H ) `
 d )  =  ( 2nd `  d
) )
6564eleq1d 2524 . . . . . . . . . . . . . . . 16  |-  ( d  e.  H  ->  (
( ( 2nd  |`  H ) `
 d )  e.  t  <->  ( 2nd `  d
)  e.  t ) )
6665adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( d  e.  H  /\  ( 1st `  d ) 
C_  ( 1st `  f
) )  ->  (
( ( 2nd  |`  H ) `
 d )  e.  t  <->  ( 2nd `  d
)  e.  t ) )
6766pm5.74i 253 . . . . . . . . . . . . . 14  |-  ( ( ( d  e.  H  /\  ( 1st `  d
)  C_  ( 1st `  f ) )  -> 
( ( 2nd  |`  H ) `
 d )  e.  t )  <->  ( (
d  e.  H  /\  ( 1st `  d ) 
C_  ( 1st `  f
) )  ->  ( 2nd `  d )  e.  t ) )
68 impexp 452 . . . . . . . . . . . . . 14  |-  ( ( ( d  e.  H  /\  ( 1st `  d
)  C_  ( 1st `  f ) )  -> 
( 2nd `  d
)  e.  t )  <-> 
( d  e.  H  ->  ( ( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t ) ) )
6967, 68bitri 257 . . . . . . . . . . . . 13  |-  ( ( ( d  e.  H  /\  ( 1st `  d
)  C_  ( 1st `  f ) )  -> 
( ( 2nd  |`  H ) `
 d )  e.  t )  <->  ( d  e.  H  ->  ( ( 1st `  d ) 
C_  ( 1st `  f
)  ->  ( 2nd `  d )  e.  t ) ) )
7063, 69syl6bb 269 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( d  e.  ( ( tail `  D
) `  f )  ->  ( ( 2nd  |`  H ) `
 d )  e.  t )  <->  ( d  e.  H  ->  ( ( 1st `  d ) 
C_  ( 1st `  f
)  ->  ( 2nd `  d )  e.  t ) ) ) )
7170ralbidv2 2835 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  ( A. d  e.  (
( tail `  D ) `  f ) ( ( 2nd  |`  H ) `  d )  e.  t  <->  A. d  e.  H  ( ( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t ) ) )
7249, 71bitrd 261 . . . . . . . . . 10  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  f  e.  H )  ->  (
( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t  <->  A. d  e.  H  ( ( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t ) ) )
7372rexbidva 2910 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( E. f  e.  H  ( ( 2nd  |`  H )
" ( ( tail `  D ) `  f
) )  C_  t  <->  E. f  e.  H  A. d  e.  H  (
( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t ) ) )
74 vex 3060 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
75 vex 3060 . . . . . . . . . . . . . . . . 17  |-  v  e. 
_V
7674, 75op1std 6835 . . . . . . . . . . . . . . . 16  |-  ( d  =  <. k ,  v
>.  ->  ( 1st `  d
)  =  k )
7776sseq1d 3471 . . . . . . . . . . . . . . 15  |-  ( d  =  <. k ,  v
>.  ->  ( ( 1st `  d )  C_  ( 1st `  f )  <->  k  C_  ( 1st `  f ) ) )
7874, 75op2ndd 6836 . . . . . . . . . . . . . . . 16  |-  ( d  =  <. k ,  v
>.  ->  ( 2nd `  d
)  =  v )
7978eleq1d 2524 . . . . . . . . . . . . . . 15  |-  ( d  =  <. k ,  v
>.  ->  ( ( 2nd `  d )  e.  t  <-> 
v  e.  t ) )
8077, 79imbi12d 326 . . . . . . . . . . . . . 14  |-  ( d  =  <. k ,  v
>.  ->  ( ( ( 1st `  d ) 
C_  ( 1st `  f
)  ->  ( 2nd `  d )  e.  t )  <->  ( k  C_  ( 1st `  f )  ->  v  e.  t ) ) )
8180raliunxp 4996 . . . . . . . . . . . . 13  |-  ( A. d  e.  U_  k  e.  F  ( { k }  X.  k ) ( ( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  A. k  e.  F  A. v  e.  k  ( k  C_  ( 1st `  f
)  ->  v  e.  t ) )
82 sneq 3990 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  { n }  =  { k } )
83 id 22 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  n  =  k )
8482, 83xpeq12d 4881 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  ( { n }  X.  n )  =  ( { k }  X.  k ) )
8584cbviunv 4331 . . . . . . . . . . . . . . 15  |-  U_ n  e.  F  ( {
n }  X.  n
)  =  U_ k  e.  F  ( {
k }  X.  k
)
861, 85eqtri 2484 . . . . . . . . . . . . . 14  |-  H  = 
U_ k  e.  F  ( { k }  X.  k )
8786raleqi 3003 . . . . . . . . . . . . 13  |-  ( A. d  e.  H  (
( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  A. d  e.  U_  k  e.  F  ( { k }  X.  k ) ( ( 1st `  d ) 
C_  ( 1st `  f
)  ->  ( 2nd `  d )  e.  t ) )
88 dfss3 3434 . . . . . . . . . . . . . . . 16  |-  ( k 
C_  t  <->  A. v  e.  k  v  e.  t )
8988imbi2i 318 . . . . . . . . . . . . . . 15  |-  ( ( k  C_  ( 1st `  f )  ->  k  C_  t )  <->  ( k  C_  ( 1st `  f
)  ->  A. v  e.  k  v  e.  t ) )
90 r19.21v 2805 . . . . . . . . . . . . . . 15  |-  ( A. v  e.  k  (
k  C_  ( 1st `  f )  ->  v  e.  t )  <->  ( k  C_  ( 1st `  f
)  ->  A. v  e.  k  v  e.  t ) )
9189, 90bitr4i 260 . . . . . . . . . . . . . 14  |-  ( ( k  C_  ( 1st `  f )  ->  k  C_  t )  <->  A. v  e.  k  ( k  C_  ( 1st `  f
)  ->  v  e.  t ) )
9291ralbii 2831 . . . . . . . . . . . . 13  |-  ( A. k  e.  F  (
k  C_  ( 1st `  f )  ->  k  C_  t )  <->  A. k  e.  F  A. v  e.  k  ( k  C_  ( 1st `  f
)  ->  v  e.  t ) )
9381, 87, 923bitr4i 285 . . . . . . . . . . . 12  |-  ( A. d  e.  H  (
( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  A. k  e.  F  ( k  C_  ( 1st `  f
)  ->  k  C_  t ) )
9493rexbii 2901 . . . . . . . . . . 11  |-  ( E. f  e.  H  A. d  e.  H  (
( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  E. f  e.  H  A. k  e.  F  ( k  C_  ( 1st `  f
)  ->  k  C_  t ) )
951rexeqi 3004 . . . . . . . . . . 11  |-  ( E. f  e.  H  A. k  e.  F  (
k  C_  ( 1st `  f )  ->  k  C_  t )  <->  E. f  e.  U_  n  e.  F  ( { n }  X.  n ) A. k  e.  F  ( k  C_  ( 1st `  f
)  ->  k  C_  t ) )
96 vex 3060 . . . . . . . . . . . . . . . 16  |-  n  e. 
_V
97 vex 3060 . . . . . . . . . . . . . . . 16  |-  m  e. 
_V
9896, 97op1std 6835 . . . . . . . . . . . . . . 15  |-  ( f  =  <. n ,  m >.  ->  ( 1st `  f
)  =  n )
9998sseq2d 3472 . . . . . . . . . . . . . 14  |-  ( f  =  <. n ,  m >.  ->  ( k  C_  ( 1st `  f )  <-> 
k  C_  n )
)
10099imbi1d 323 . . . . . . . . . . . . 13  |-  ( f  =  <. n ,  m >.  ->  ( ( k 
C_  ( 1st `  f
)  ->  k  C_  t )  <->  ( k  C_  n  ->  k  C_  t ) ) )
101100ralbidv 2839 . . . . . . . . . . . 12  |-  ( f  =  <. n ,  m >.  ->  ( A. k  e.  F  ( k  C_  ( 1st `  f
)  ->  k  C_  t )  <->  A. k  e.  F  ( k  C_  n  ->  k  C_  t ) ) )
102101rexiunxp 4997 . . . . . . . . . . 11  |-  ( E. f  e.  U_  n  e.  F  ( {
n }  X.  n
) A. k  e.  F  ( k  C_  ( 1st `  f )  ->  k  C_  t
)  <->  E. n  e.  F  E. m  e.  n  A. k  e.  F  ( k  C_  n  ->  k  C_  t )
)
10394, 95, 1023bitri 279 . . . . . . . . . 10  |-  ( E. f  e.  H  A. d  e.  H  (
( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  E. n  e.  F  E. m  e.  n  A. k  e.  F  ( k  C_  n  ->  k  C_  t ) )
104 fileln0 20920 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  n  =/=  (/) )
105104adantlr 726 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  n  e.  F )  ->  n  =/=  (/) )
106 r19.9rzv 3875 . . . . . . . . . . . . 13  |-  ( n  =/=  (/)  ->  ( A. k  e.  F  (
k  C_  n  ->  k 
C_  t )  <->  E. m  e.  n  A. k  e.  F  ( k  C_  n  ->  k  C_  t ) ) )
107105, 106syl 17 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  n  e.  F )  ->  ( A. k  e.  F  ( k  C_  n  ->  k  C_  t )  <->  E. m  e.  n  A. k  e.  F  (
k  C_  n  ->  k 
C_  t ) ) )
108 ssid 3463 . . . . . . . . . . . . . . 15  |-  n  C_  n
109 sseq1 3465 . . . . . . . . . . . . . . . . 17  |-  ( k  =  n  ->  (
k  C_  n  <->  n  C_  n
) )
110 sseq1 3465 . . . . . . . . . . . . . . . . 17  |-  ( k  =  n  ->  (
k  C_  t  <->  n  C_  t
) )
111109, 110imbi12d 326 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  (
( k  C_  n  ->  k  C_  t )  <->  ( n  C_  n  ->  n 
C_  t ) ) )
112111rspcv 3158 . . . . . . . . . . . . . . 15  |-  ( n  e.  F  ->  ( A. k  e.  F  ( k  C_  n  ->  k  C_  t )  ->  ( n  C_  n  ->  n  C_  t )
) )
113108, 112mpii 44 . . . . . . . . . . . . . 14  |-  ( n  e.  F  ->  ( A. k  e.  F  ( k  C_  n  ->  k  C_  t )  ->  n  C_  t )
)
114113adantl 472 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  n  e.  F )  ->  ( A. k  e.  F  ( k  C_  n  ->  k  C_  t )  ->  n  C_  t )
)
115 sstr2 3451 . . . . . . . . . . . . . . 15  |-  ( k 
C_  n  ->  (
n  C_  t  ->  k 
C_  t ) )
116115com12 32 . . . . . . . . . . . . . 14  |-  ( n 
C_  t  ->  (
k  C_  n  ->  k 
C_  t ) )
117116ralrimivw 2815 . . . . . . . . . . . . 13  |-  ( n 
C_  t  ->  A. k  e.  F  ( k  C_  n  ->  k  C_  t ) )
118114, 117impbid1 208 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  n  e.  F )  ->  ( A. k  e.  F  ( k  C_  n  ->  k  C_  t )  <->  n 
C_  t ) )
119107, 118bitr3d 263 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( Fil `  X )  /\  t  C_  X
)  /\  n  e.  F )  ->  ( E. m  e.  n  A. k  e.  F  ( k  C_  n  ->  k  C_  t )  <->  n 
C_  t ) )
120119rexbidva 2910 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( E. n  e.  F  E. m  e.  n  A. k  e.  F  ( k  C_  n  ->  k  C_  t )  <->  E. n  e.  F  n 
C_  t ) )
121103, 120syl5bb 265 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( E. f  e.  H  A. d  e.  H  ( ( 1st `  d
)  C_  ( 1st `  f )  ->  ( 2nd `  d )  e.  t )  <->  E. n  e.  F  n  C_  t
) )
12232, 73, 1213bitrd 287 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  t  C_  X )  ->  ( E. d  e.  ran  ( tail `  D )
( ( 2nd  |`  H )
" d )  C_  t 
<->  E. n  e.  F  n  C_  t ) )
123122pm5.32da 651 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( (
t  C_  X  /\  E. d  e.  ran  ( tail `  D ) ( ( 2nd  |`  H )
" d )  C_  t )  <->  ( t  C_  X  /\  E. n  e.  F  n  C_  t
) ) )
124 filn0 20932 . . . . . . . . . . . 12  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
12596snnz 4103 . . . . . . . . . . . . . . . 16  |-  { n }  =/=  (/)
126104, 125jctil 544 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  ( { n }  =/=  (/) 
/\  n  =/=  (/) ) )
127 neanior 2728 . . . . . . . . . . . . . . 15  |-  ( ( { n }  =/=  (/) 
/\  n  =/=  (/) )  <->  -.  ( { n }  =  (/) 
\/  n  =  (/) ) )
128126, 127sylib 201 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  -.  ( { n }  =  (/) 
\/  n  =  (/) ) )
129 ss0b 3776 . . . . . . . . . . . . . . 15  |-  ( ( { n }  X.  n )  C_  (/)  <->  ( {
n }  X.  n
)  =  (/) )
130 xpeq0 5279 . . . . . . . . . . . . . . 15  |-  ( ( { n }  X.  n )  =  (/)  <->  ( { n }  =  (/) 
\/  n  =  (/) ) )
131129, 130bitri 257 . . . . . . . . . . . . . 14  |-  ( ( { n }  X.  n )  C_  (/)  <->  ( {
n }  =  (/)  \/  n  =  (/) ) )
132128, 131sylnibr 311 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  -.  ( { n }  X.  n )  C_  (/) )
133132ralrimiva 2814 . . . . . . . . . . . 12  |-  ( F  e.  ( Fil `  X
)  ->  A. n  e.  F  -.  ( { n }  X.  n )  C_  (/) )
134 r19.2z 3870 . . . . . . . . . . . 12  |-  ( ( F  =/=  (/)  /\  A. n  e.  F  -.  ( { n }  X.  n )  C_  (/) )  ->  E. n  e.  F  -.  ( { n }  X.  n )  C_  (/) )
135124, 133, 134syl2anc 671 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  E. n  e.  F  -.  ( { n }  X.  n )  C_  (/) )
136 rexnal 2848 . . . . . . . . . . 11  |-  ( E. n  e.  F  -.  ( { n }  X.  n )  C_  (/)  <->  -.  A. n  e.  F  ( {
n }  X.  n
)  C_  (/) )
137135, 136sylib 201 . . . . . . . . . 10  |-  ( F  e.  ( Fil `  X
)  ->  -.  A. n  e.  F  ( {
n }  X.  n
)  C_  (/) )
1381sseq1i 3468 . . . . . . . . . . . 12  |-  ( H 
C_  (/)  <->  U_ n  e.  F  ( { n }  X.  n )  C_  (/) )
139 ss0b 3776 . . . . . . . . . . . 12  |-  ( H 
C_  (/)  <->  H  =  (/) )
140 iunss 4333 . . . . . . . . . . . 12  |-  ( U_ n  e.  F  ( { n }  X.  n )  C_  (/)  <->  A. n  e.  F  ( {
n }  X.  n
)  C_  (/) )
141138, 139, 1403bitr3i 283 . . . . . . . . . . 11  |-  ( H  =  (/)  <->  A. n  e.  F  ( { n }  X.  n )  C_  (/) )
142141necon3abii 2682 . . . . . . . . . 10  |-  ( H  =/=  (/)  <->  -.  A. n  e.  F  ( {
n }  X.  n
)  C_  (/) )
143137, 142sylibr 217 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  H  =/=  (/) )
144 dmresi 5182 . . . . . . . . . . . 12  |-  dom  (  _I  |`  H )  =  H
1451, 2filnetlem2 31085 . . . . . . . . . . . . . 14  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
146145simpli 464 . . . . . . . . . . . . 13  |-  (  _I  |`  H )  C_  D
147 dmss 5056 . . . . . . . . . . . . 13  |-  ( (  _I  |`  H )  C_  D  ->  dom  (  _I  |`  H )  C_  dom  D )
148146, 147ax-mp 5 . . . . . . . . . . . 12  |-  dom  (  _I  |`  H )  C_  dom  D
149144, 148eqsstr3i 3475 . . . . . . . . . . 11  |-  H  C_  dom  D
150145simpri 468 . . . . . . . . . . . . 13  |-  D  C_  ( H  X.  H
)
151 dmss 5056 . . . . . . . . . . . . 13  |-  ( D 
C_  ( H  X.  H )  ->  dom  D 
C_  dom  ( H  X.  H ) )
152150, 151ax-mp 5 . . . . . . . . . . . 12  |-  dom  D  C_ 
dom  ( H  X.  H )
153 dmxpid 5076 . . . . . . . . . . . 12  |-  dom  ( H  X.  H )  =  H
154152, 153sseqtri 3476 . . . . . . . . . . 11  |-  dom  D  C_  H
155149, 154eqssi 3460 . . . . . . . . . 10  |-  H  =  dom  D
156155tailfb 31083 . . . . . . . . 9  |-  ( ( D  e.  DirRel  /\  H  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  H
) )
1575, 143, 156syl2anc 671 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ran  ( tail `  D )  e.  (
fBas `  H )
)
158 elfm 21017 . . . . . . . 8  |-  ( ( X  e.  F  /\  ran  ( tail `  D
)  e.  ( fBas `  H )  /\  ( 2nd  |`  H ) : H --> X )  -> 
( t  e.  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D
) )  <->  ( t  C_  X  /\  E. d  e.  ran  ( tail `  D
) ( ( 2nd  |`  H ) " d
)  C_  t )
) )
15910, 157, 9, 158syl3anc 1276 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  ( 2nd  |`  H )
) `  ran  ( tail `  D ) )  <->  ( t  C_  X  /\  E. d  e.  ran  ( tail `  D
) ( ( 2nd  |`  H ) " d
)  C_  t )
) )
160 filfbas 20918 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
161 elfg 20941 . . . . . . . 8  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. n  e.  F  n  C_  t ) ) )
162160, 161syl 17 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. n  e.  F  n  C_  t ) ) )
163123, 159, 1623bitr4d 293 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  ( 2nd  |`  H )
) `  ran  ( tail `  D ) )  <->  t  e.  ( X filGen F ) ) )
164163eqrdv 2460 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) )  =  ( X filGen F ) )
165 fgfil 20945 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
166164, 165eqtr2d 2497 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  =  ( ( X  FilMap  ( 2nd  |`  H )
) `  ran  ( tail `  D ) ) )
16721, 166jca 539 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( ( 2nd  |`  H ) : dom  D --> X  /\  F  =  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) ) ) )
168 feq1 5736 . . . . 5  |-  ( f  =  ( 2nd  |`  H )  ->  ( f : dom  D --> X  <->  ( 2nd  |`  H ) : dom  D --> X ) )
169 oveq2 6328 . . . . . . 7  |-  ( f  =  ( 2nd  |`  H )  ->  ( X  FilMap  f )  =  ( X 
FilMap  ( 2nd  |`  H ) ) )
170169fveq1d 5894 . . . . . 6  |-  ( f  =  ( 2nd  |`  H )  ->  ( ( X 
FilMap  f ) `  ran  ( tail `  D )
)  =  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) ) )
171170eqeq2d 2472 . . . . 5  |-  ( f  =  ( 2nd  |`  H )  ->  ( F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) )  <-> 
F  =  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) ) ) )
172168, 171anbi12d 722 . . . 4  |-  ( f  =  ( 2nd  |`  H )  ->  ( ( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D
) ) )  <->  ( ( 2nd  |`  H ) : dom  D --> X  /\  F  =  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) ) ) ) )
173172spcegv 3147 . . 3  |-  ( ( 2nd  |`  H )  e.  _V  ->  ( (
( 2nd  |`  H ) : dom  D --> X  /\  F  =  ( ( X  FilMap  ( 2nd  |`  H ) ) `  ran  ( tail `  D ) ) )  ->  E. f
( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) ) ) ) )
17415, 167, 173sylc 62 . 2  |-  ( F  e.  ( Fil `  X
)  ->  E. f
( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) ) ) )
175 dmeq 5057 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
176175feq2d 5741 . . . . 5  |-  ( d  =  D  ->  (
f : dom  d --> X 
<->  f : dom  D --> X ) )
177 fveq2 5892 . . . . . . . 8  |-  ( d  =  D  ->  ( tail `  d )  =  ( tail `  D
) )
178177rneqd 5084 . . . . . . 7  |-  ( d  =  D  ->  ran  ( tail `  d )  =  ran  ( tail `  D
) )
179178fveq2d 5896 . . . . . 6  |-  ( d  =  D  ->  (
( X  FilMap  f ) `
 ran  ( tail `  d ) )  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) ) )
180179eqeq2d 2472 . . . . 5  |-  ( d  =  D  ->  ( F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d
) )  <->  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) ) ) )
181176, 180anbi12d 722 . . . 4  |-  ( d  =  D  ->  (
( f : dom  d
--> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d ) ) )  <->  ( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D
) ) ) ) )
182181exbidv 1779 . . 3  |-  ( d  =  D  ->  ( E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d
) ) )  <->  E. f
( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D ) ) ) ) )
183182rspcev 3162 . 2  |-  ( ( D  e.  DirRel  /\  E. f ( f : dom  D --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  D
) ) ) )  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d
) ) ) )
1845, 174, 183syl2anc 671 1  |-  ( F  e.  ( Fil `  X
)  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `
 ran  ( tail `  d ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {csn 3980   <.cop 3986   U.cuni 4212   U_ciun 4292   class class class wbr 4418   {copab 4476    _I cid 4766    X. cxp 4854   dom cdm 4856   ran crn 4857    |` cres 4858   "cima 4859   Fun wfun 5599    Fn wfn 5600   -->wf 5601   -onto->wfo 5603   ` cfv 5605  (class class class)co 6320   1stc1st 6823   2ndc2nd 6824   DirRelcdir 16529   tailctail 16530   fBascfbas 19013   filGencfg 19014   Filcfil 20915    FilMap cfm 21003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-1st 6825  df-2nd 6826  df-dir 16531  df-tail 16532  df-fbas 19022  df-fg 19023  df-fil 20916  df-fm 21008
This theorem is referenced by:  filnet  31088
  Copyright terms: Public domain W3C validator