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Theorem filnetlem3 26329
Description: Lemma for filnet 26331. (Contributed by Jeff Hankins, 13-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
filnetlem3  |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) ) )
Distinct variable groups:    x, y, n, F    x, H, y   
n, X
Allowed substitution hints:    D( x, y, n)    H( n)    X( x, y)

Proof of Theorem filnetlem3
Dummy variables  u  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmresi 5005 . . . . . 6  |-  dom  (  _I  |`  H )  =  H
2 filnet.h . . . . . . . . 9  |-  H  = 
U_ n  e.  F  ( { n }  X.  n )
3 filnet.d . . . . . . . . 9  |-  D  =  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  ( 1st `  y
)  C_  ( 1st `  x ) ) }
42, 3filnetlem2 26328 . . . . . . . 8  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
54simpli 444 . . . . . . 7  |-  (  _I  |`  H )  C_  D
6 dmss 4878 . . . . . . 7  |-  ( (  _I  |`  H )  C_  D  ->  dom  (  _I  |`  H )  C_  dom  D )
75, 6ax-mp 8 . . . . . 6  |-  dom  (  _I  |`  H )  C_  dom  D
81, 7eqsstr3i 3209 . . . . 5  |-  H  C_  dom  D
9 ssun1 3338 . . . . 5  |-  dom  D  C_  ( dom  D  u.  ran  D )
108, 9sstri 3188 . . . 4  |-  H  C_  ( dom  D  u.  ran  D )
11 dmrnssfld 4938 . . . 4  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
1210, 11sstri 3188 . . 3  |-  H  C_  U.
U. D
134simpri 448 . . . . 5  |-  D  C_  ( H  X.  H
)
14 uniss 3848 . . . . 5  |-  ( D 
C_  ( H  X.  H )  ->  U. D  C_ 
U. ( H  X.  H ) )
15 uniss 3848 . . . . 5  |-  ( U. D  C_  U. ( H  X.  H )  ->  U. U. D  C_  U. U. ( H  X.  H
) )
1613, 14, 15mp2b 9 . . . 4  |-  U. U. D  C_  U. U. ( H  X.  H )
17 unixpss 4799 . . . . 5  |-  U. U. ( H  X.  H
)  C_  ( H  u.  H )
18 unidm 3318 . . . . 5  |-  ( H  u.  H )  =  H
1917, 18sseqtri 3210 . . . 4  |-  U. U. ( H  X.  H
)  C_  H
2016, 19sstri 3188 . . 3  |-  U. U. D  C_  H
2112, 20eqssi 3195 . 2  |-  H  = 
U. U. D
22 filelss 17547 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  n  C_  X )
23 xpss2 4796 . . . . . . . 8  |-  ( n 
C_  X  ->  ( { n }  X.  n )  C_  ( { n }  X.  X ) )
2422, 23syl 15 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  ( { n }  X.  n )  C_  ( { n }  X.  X ) )
2524ralrimiva 2626 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  A. n  e.  F  ( {
n }  X.  n
)  C_  ( {
n }  X.  X
) )
26 ss2iun 3920 . . . . . 6  |-  ( A. n  e.  F  ( { n }  X.  n )  C_  ( { n }  X.  X )  ->  U_ n  e.  F  ( {
n }  X.  n
)  C_  U_ n  e.  F  ( { n }  X.  X ) )
2725, 26syl 15 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  U_ n  e.  F  ( { n }  X.  n )  C_  U_ n  e.  F  ( { n }  X.  X ) )
28 iunxpconst 4746 . . . . 5  |-  U_ n  e.  F  ( {
n }  X.  X
)  =  ( F  X.  X )
2927, 28syl6sseq 3224 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  U_ n  e.  F  ( { n }  X.  n )  C_  ( F  X.  X
) )
302, 29syl5eqss 3222 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  H  C_  ( F  X.  X ) )
315a1i 10 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  (  _I  |`  H )  C_  D
)
323relopabi 4811 . . . . 5  |-  Rel  D
3331, 32jctil 523 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( Rel  D  /\  (  _I  |`  H ) 
C_  D ) )
34 simpl 443 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  F  e.  ( Fil `  X ) )
3530adantr 451 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  H  C_  ( F  X.  X ) )
36 simprl 732 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  v  e.  H )
3735, 36sseldd 3181 . . . . . . . . . . 11  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  v  e.  ( F  X.  X
) )
38 xp1st 6149 . . . . . . . . . . 11  |-  ( v  e.  ( F  X.  X )  ->  ( 1st `  v )  e.  F )
3937, 38syl 15 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( 1st `  v )  e.  F
)
40 simprr 733 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  z  e.  H )
4135, 40sseldd 3181 . . . . . . . . . . 11  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  z  e.  ( F  X.  X
) )
42 xp1st 6149 . . . . . . . . . . 11  |-  ( z  e.  ( F  X.  X )  ->  ( 1st `  z )  e.  F )
4341, 42syl 15 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( 1st `  z )  e.  F
)
44 filinn0 17555 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  ( 1st `  v )  e.  F  /\  ( 1st `  z )  e.  F
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  =/=  (/) )
4534, 39, 43, 44syl3anc 1182 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  =/=  (/) )
46 n0 3464 . . . . . . . . 9  |-  ( ( ( 1st `  v
)  i^i  ( 1st `  z ) )  =/=  (/) 
<->  E. u  u  e.  ( ( 1st `  v
)  i^i  ( 1st `  z ) ) )
4745, 46sylib 188 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  E. u  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )
4836adantr 451 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
v  e.  H )
49 filin 17549 . . . . . . . . . . . . . . . 16  |-  ( ( F  e.  ( Fil `  X )  /\  ( 1st `  v )  e.  F  /\  ( 1st `  z )  e.  F
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  F
)
5034, 39, 43, 49syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  F
)
5150adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
( ( 1st `  v
)  i^i  ( 1st `  z ) )  e.  F )
52 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  ->  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )
53 id 19 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 1st `  v )  i^i  ( 1st `  z ) )  ->  n  =  ( ( 1st `  v
)  i^i  ( 1st `  z ) ) )
5453opeliunxp2 4824 . . . . . . . . . . . . . 14  |-  ( <.
( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  U_ n  e.  F  ( {
n }  X.  n
)  <->  ( ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  F  /\  u  e.  (
( 1st `  v
)  i^i  ( 1st `  z ) ) ) )
5551, 52, 54sylanbrc 645 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  ->  <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  U_ n  e.  F  ( {
n }  X.  n
) )
5655, 2syl6eleqr 2374 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  ->  <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
)
57 fvex 5539 . . . . . . . . . . . . . . . 16  |-  ( 1st `  v )  e.  _V
5857inex1 4155 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  _V
59 vex 2791 . . . . . . . . . . . . . . 15  |-  u  e. 
_V
6058, 59op1st 6128 . . . . . . . . . . . . . 14  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  =  ( ( 1st `  v )  i^i  ( 1st `  z ) )
61 inss1 3389 . . . . . . . . . . . . . 14  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  C_  ( 1st `  v )
6260, 61eqsstri 3208 . . . . . . . . . . . . 13  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  v
)
63 vex 2791 . . . . . . . . . . . . . 14  |-  v  e. 
_V
64 opex 4237 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  _V
652, 3, 63, 64filnetlem1 26327 . . . . . . . . . . . . 13  |-  ( v D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( (
v  e.  H  /\  <.
( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
)  /\  ( 1st ` 
<. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  v ) ) )
6662, 65mpbiran2 885 . . . . . . . . . . . 12  |-  ( v D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( v  e.  H  /\  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
) )
6748, 56, 66sylanbrc 645 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
v D <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. )
6840adantr 451 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
z  e.  H )
69 inss2 3390 . . . . . . . . . . . . . 14  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  C_  ( 1st `  z )
7060, 69eqsstri 3208 . . . . . . . . . . . . 13  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  z
)
71 vex 2791 . . . . . . . . . . . . . 14  |-  z  e. 
_V
722, 3, 71, 64filnetlem1 26327 . . . . . . . . . . . . 13  |-  ( z D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( (
z  e.  H  /\  <.
( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
)  /\  ( 1st ` 
<. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  z ) ) )
7370, 72mpbiran2 885 . . . . . . . . . . . 12  |-  ( z D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( z  e.  H  /\  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
) )
7468, 56, 73sylanbrc 645 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
z D <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. )
75 breq2 4027 . . . . . . . . . . . . 13  |-  ( w  =  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  ->  (
v D w  <->  v D <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. ) )
76 breq2 4027 . . . . . . . . . . . . 13  |-  ( w  =  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  ->  (
z D w  <->  z D <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. ) )
7775, 76anbi12d 691 . . . . . . . . . . . 12  |-  ( w  =  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  ->  (
( v D w  /\  z D w )  <->  ( v D
<. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  /\  z D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )
) )
7864, 77spcev 2875 . . . . . . . . . . 11  |-  ( ( v D <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  /\  z D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  ->  E. w ( v D w  /\  z D w ) )
7967, 74, 78syl2anc 642 . . . . . . . . . 10  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  ->  E. w ( v D w  /\  z D w ) )
8079ex 423 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( u  e.  ( ( 1st `  v
)  i^i  ( 1st `  z ) )  ->  E. w ( v D w  /\  z D w ) ) )
8180exlimdv 1664 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( E. u  u  e.  (
( 1st `  v
)  i^i  ( 1st `  z ) )  ->  E. w ( v D w  /\  z D w ) ) )
8247, 81mpd 14 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  E. w
( v D w  /\  z D w ) )
8382ralrimivva 2635 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  A. v  e.  H  A. z  e.  H  E. w
( v D w  /\  z D w ) )
84 codir 5063 . . . . . 6  |-  ( ( H  X.  H ) 
C_  ( `' D  o.  D )  <->  A. v  e.  H  A. z  e.  H  E. w
( v D w  /\  z D w ) )
8583, 84sylibr 203 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( H  X.  H )  C_  ( `' D  o.  D
) )
86 vex 2791 . . . . . . . . . . . . 13  |-  w  e. 
_V
872, 3, 63, 86filnetlem1 26327 . . . . . . . . . . . 12  |-  ( v D w  <->  ( (
v  e.  H  /\  w  e.  H )  /\  ( 1st `  w
)  C_  ( 1st `  v ) ) )
8887simplbi 446 . . . . . . . . . . 11  |-  ( v D w  ->  (
v  e.  H  /\  w  e.  H )
)
8988simpld 445 . . . . . . . . . 10  |-  ( v D w  ->  v  e.  H )
902, 3, 86, 71filnetlem1 26327 . . . . . . . . . . . 12  |-  ( w D z  <->  ( (
w  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  w ) ) )
9190simplbi 446 . . . . . . . . . . 11  |-  ( w D z  ->  (
w  e.  H  /\  z  e.  H )
)
9291simprd 449 . . . . . . . . . 10  |-  ( w D z  ->  z  e.  H )
9389, 92anim12i 549 . . . . . . . . 9  |-  ( ( v D w  /\  w D z )  -> 
( v  e.  H  /\  z  e.  H
) )
9490simprbi 450 . . . . . . . . . 10  |-  ( w D z  ->  ( 1st `  z )  C_  ( 1st `  w ) )
9587simprbi 450 . . . . . . . . . 10  |-  ( v D w  ->  ( 1st `  w )  C_  ( 1st `  v ) )
9694, 95sylan9ssr 3193 . . . . . . . . 9  |-  ( ( v D w  /\  w D z )  -> 
( 1st `  z
)  C_  ( 1st `  v ) )
972, 3, 63, 71filnetlem1 26327 . . . . . . . . 9  |-  ( v D z  <->  ( (
v  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  v ) ) )
9893, 96, 97sylanbrc 645 . . . . . . . 8  |-  ( ( v D w  /\  w D z )  -> 
v D z )
9998ax-gen 1533 . . . . . . 7  |-  A. z
( ( v D w  /\  w D z )  ->  v D z )
10099gen2 1534 . . . . . 6  |-  A. v A. w A. z ( ( v D w  /\  w D z )  ->  v D
z )
101 cotr 5055 . . . . . 6  |-  ( ( D  o.  D ) 
C_  D  <->  A. v A. w A. z ( ( v D w  /\  w D z )  ->  v D
z ) )
102100, 101mpbir 200 . . . . 5  |-  ( D  o.  D )  C_  D
10385, 102jctil 523 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( ( D  o.  D )  C_  D  /\  ( H  X.  H )  C_  ( `' D  o.  D
) ) )
104 filtop 17550 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
105 xpexg 4800 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  X  e.  F )  ->  ( F  X.  X )  e. 
_V )
106104, 105mpdan 649 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( F  X.  X )  e.  _V )
107 ssexg 4160 . . . . . . . 8  |-  ( ( H  C_  ( F  X.  X )  /\  ( F  X.  X )  e. 
_V )  ->  H  e.  _V )
10830, 106, 107syl2anc 642 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  H  e.  _V )
109 xpexg 4800 . . . . . . 7  |-  ( ( H  e.  _V  /\  H  e.  _V )  ->  ( H  X.  H
)  e.  _V )
110108, 108, 109syl2anc 642 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( H  X.  H )  e.  _V )
111 ssexg 4160 . . . . . 6  |-  ( ( D  C_  ( H  X.  H )  /\  ( H  X.  H )  e. 
_V )  ->  D  e.  _V )
11213, 110, 111sylancr 644 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  D  e.  _V )
11321isdir 14354 . . . . 5  |-  ( D  e.  _V  ->  ( D  e.  DirRel  <->  ( ( Rel  D  /\  (  _I  |`  H )  C_  D
)  /\  ( ( D  o.  D )  C_  D  /\  ( H  X.  H )  C_  ( `' D  o.  D
) ) ) ) )
114112, 113syl 15 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( D  e.  DirRel 
<->  ( ( Rel  D  /\  (  _I  |`  H ) 
C_  D )  /\  ( ( D  o.  D )  C_  D  /\  ( H  X.  H
)  C_  ( `' D  o.  D )
) ) ) )
11533, 103, 114mpbir2and 888 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  D  e.  DirRel )
11630, 115jca 518 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) )
11721, 116pm3.2i 441 1  |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023   {copab 4076    _I cid 4304    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   ` cfv 5255   1stc1st 6120   DirRelcdir 14350   Filcfil 17540
This theorem is referenced by:  filnetlem4  26330
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-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-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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-1st 6122  df-dir 14352  df-fbas 17520  df-fil 17541
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