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Theorem filnetlem3 30438
Description: Lemma for filnet 30440. (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 5317 . . . . . 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 30437 . . . . . . . 8  |-  ( (  _I  |`  H )  C_  D  /\  D  C_  ( H  X.  H
) )
54simpli 456 . . . . . . 7  |-  (  _I  |`  H )  C_  D
6 dmss 5191 . . . . . . 7  |-  ( (  _I  |`  H )  C_  D  ->  dom  (  _I  |`  H )  C_  dom  D )
75, 6ax-mp 5 . . . . . 6  |-  dom  (  _I  |`  H )  C_  dom  D
81, 7eqsstr3i 3520 . . . . 5  |-  H  C_  dom  D
9 ssun1 3653 . . . . 5  |-  dom  D  C_  ( dom  D  u.  ran  D )
108, 9sstri 3498 . . . 4  |-  H  C_  ( dom  D  u.  ran  D )
11 dmrnssfld 5250 . . . 4  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
1210, 11sstri 3498 . . 3  |-  H  C_  U.
U. D
134simpri 460 . . . . 5  |-  D  C_  ( H  X.  H
)
14 uniss 4256 . . . . 5  |-  ( D 
C_  ( H  X.  H )  ->  U. D  C_ 
U. ( H  X.  H ) )
15 uniss 4256 . . . . 5  |-  ( U. D  C_  U. ( H  X.  H )  ->  U. U. D  C_  U. U. ( H  X.  H
) )
1613, 14, 15mp2b 10 . . . 4  |-  U. U. D  C_  U. U. ( H  X.  H )
17 unixpss 5106 . . . . 5  |-  U. U. ( H  X.  H
)  C_  ( H  u.  H )
18 unidm 3633 . . . . 5  |-  ( H  u.  H )  =  H
1917, 18sseqtri 3521 . . . 4  |-  U. U. ( H  X.  H
)  C_  H
2016, 19sstri 3498 . . 3  |-  U. U. D  C_  H
2112, 20eqssi 3505 . 2  |-  H  = 
U. U. D
22 filelss 20519 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  n  C_  X )
23 xpss2 5100 . . . . . . . 8  |-  ( n 
C_  X  ->  ( { n }  X.  n )  C_  ( { n }  X.  X ) )
2422, 23syl 16 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  n  e.  F )  ->  ( { n }  X.  n )  C_  ( { n }  X.  X ) )
2524ralrimiva 2868 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  A. n  e.  F  ( {
n }  X.  n
)  C_  ( {
n }  X.  X
) )
26 ss2iun 4331 . . . . . 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 16 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  U_ n  e.  F  ( { n }  X.  n )  C_  U_ n  e.  F  ( { n }  X.  X ) )
28 iunxpconst 5045 . . . . 5  |-  U_ n  e.  F  ( {
n }  X.  X
)  =  ( F  X.  X )
2927, 28syl6sseq 3535 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  U_ n  e.  F  ( { n }  X.  n )  C_  ( F  X.  X
) )
302, 29syl5eqss 3533 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  H  C_  ( F  X.  X ) )
315a1i 11 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  (  _I  |`  H )  C_  D
)
323relopabi 5116 . . . . 5  |-  Rel  D
3331, 32jctil 535 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( Rel  D  /\  (  _I  |`  H ) 
C_  D ) )
34 simpl 455 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  F  e.  ( Fil `  X ) )
3530adantr 463 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  H  C_  ( F  X.  X ) )
36 simprl 754 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  v  e.  H )
3735, 36sseldd 3490 . . . . . . . . . . 11  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  v  e.  ( F  X.  X
) )
38 xp1st 6803 . . . . . . . . . . 11  |-  ( v  e.  ( F  X.  X )  ->  ( 1st `  v )  e.  F )
3937, 38syl 16 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( 1st `  v )  e.  F
)
40 simprr 755 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  z  e.  H )
4135, 40sseldd 3490 . . . . . . . . . . 11  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  z  e.  ( F  X.  X
) )
42 xp1st 6803 . . . . . . . . . . 11  |-  ( z  e.  ( F  X.  X )  ->  ( 1st `  z )  e.  F )
4341, 42syl 16 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( 1st `  z )  e.  F
)
44 filinn0 20527 . . . . . . . . . 10  |-  ( ( F  e.  ( Fil `  X )  /\  ( 1st `  v )  e.  F  /\  ( 1st `  z )  e.  F
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  =/=  (/) )
4534, 39, 43, 44syl3anc 1226 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  =/=  (/) )
46 n0 3793 . . . . . . . . 9  |-  ( ( ( 1st `  v
)  i^i  ( 1st `  z ) )  =/=  (/) 
<->  E. u  u  e.  ( ( 1st `  v
)  i^i  ( 1st `  z ) ) )
4745, 46sylib 196 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  E. u  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )
4836adantr 463 . . . . . . . . . 10  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
v  e.  H )
49 filin 20521 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( Fil `  X )  /\  ( 1st `  v )  e.  F  /\  ( 1st `  z )  e.  F
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  F
)
5034, 39, 43, 49syl3anc 1226 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  F
)
5150adantr 463 . . . . . . . . . . . 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 ) )  e.  F )
52 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( 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 22 . . . . . . . . . . . . 13  |-  ( n  =  ( ( 1st `  v )  i^i  ( 1st `  z ) )  ->  n  =  ( ( 1st `  v
)  i^i  ( 1st `  z ) ) )
5453opeliunxp2 5130 . . . . . . . . . . . 12  |-  ( <.
( ( 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 662 . . . . . . . . . . 11  |-  ( ( ( 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 2553 . . . . . . . . . 10  |-  ( ( ( 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 5858 . . . . . . . . . . . . . 14  |-  ( 1st `  v )  e.  _V
5857inex1 4578 . . . . . . . . . . . . 13  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  e.  _V
59 vex 3109 . . . . . . . . . . . . 13  |-  u  e. 
_V
6058, 59op1st 6781 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  =  ( ( 1st `  v )  i^i  ( 1st `  z ) )
61 inss1 3704 . . . . . . . . . . . 12  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  C_  ( 1st `  v )
6260, 61eqsstri 3519 . . . . . . . . . . 11  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  v
)
63 vex 3109 . . . . . . . . . . . 12  |-  v  e. 
_V
64 opex 4701 . . . . . . . . . . . 12  |-  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  _V
652, 3, 63, 64filnetlem1 30436 . . . . . . . . . . 11  |-  ( 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 917 . . . . . . . . . 10  |-  ( v D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( v  e.  H  /\  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
) )
6748, 56, 66sylanbrc 662 . . . . . . . . 9  |-  ( ( ( 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 463 . . . . . . . . . 10  |-  ( ( ( F  e.  ( Fil `  X )  /\  ( v  e.  H  /\  z  e.  H ) )  /\  u  e.  ( ( 1st `  v )  i^i  ( 1st `  z
) ) )  -> 
z  e.  H )
69 inss2 3705 . . . . . . . . . . . 12  |-  ( ( 1st `  v )  i^i  ( 1st `  z
) )  C_  ( 1st `  z )
7060, 69eqsstri 3519 . . . . . . . . . . 11  |-  ( 1st `  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >. )  C_  ( 1st `  z
)
71 vex 3109 . . . . . . . . . . . 12  |-  z  e. 
_V
722, 3, 71, 64filnetlem1 30436 . . . . . . . . . . 11  |-  ( 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 917 . . . . . . . . . 10  |-  ( z D <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  <->  ( z  e.  H  /\  <. (
( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >.  e.  H
) )
7468, 56, 73sylanbrc 662 . . . . . . . . 9  |-  ( ( ( 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 4443 . . . . . . . . . . 11  |-  ( w  =  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  ->  (
v D w  <->  v D <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. ) )
76 breq2 4443 . . . . . . . . . . 11  |-  ( w  =  <. ( ( 1st `  v )  i^i  ( 1st `  z ) ) ,  u >.  ->  (
z D w  <->  z D <. ( ( 1st `  v
)  i^i  ( 1st `  z ) ) ,  u >. ) )
7775, 76anbi12d 708 . . . . . . . . . 10  |-  ( 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 3198 . . . . . . . . 9  |-  ( ( 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 659 . . . . . . . 8  |-  ( ( ( 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 ) )
8047, 79exlimddv 1731 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  (
v  e.  H  /\  z  e.  H )
)  ->  E. w
( v D w  /\  z D w ) )
8180ralrimivva 2875 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  A. v  e.  H  A. z  e.  H  E. w
( v D w  /\  z D w ) )
82 codir 5375 . . . . . 6  |-  ( ( H  X.  H ) 
C_  ( `' D  o.  D )  <->  A. v  e.  H  A. z  e.  H  E. w
( v D w  /\  z D w ) )
8381, 82sylibr 212 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( H  X.  H )  C_  ( `' D  o.  D
) )
84 vex 3109 . . . . . . . . . . . . 13  |-  w  e. 
_V
852, 3, 63, 84filnetlem1 30436 . . . . . . . . . . . 12  |-  ( v D w  <->  ( (
v  e.  H  /\  w  e.  H )  /\  ( 1st `  w
)  C_  ( 1st `  v ) ) )
8685simplbi 458 . . . . . . . . . . 11  |-  ( v D w  ->  (
v  e.  H  /\  w  e.  H )
)
8786simpld 457 . . . . . . . . . 10  |-  ( v D w  ->  v  e.  H )
882, 3, 84, 71filnetlem1 30436 . . . . . . . . . . . 12  |-  ( w D z  <->  ( (
w  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  w ) ) )
8988simplbi 458 . . . . . . . . . . 11  |-  ( w D z  ->  (
w  e.  H  /\  z  e.  H )
)
9089simprd 461 . . . . . . . . . 10  |-  ( w D z  ->  z  e.  H )
9187, 90anim12i 564 . . . . . . . . 9  |-  ( ( v D w  /\  w D z )  -> 
( v  e.  H  /\  z  e.  H
) )
9288simprbi 462 . . . . . . . . . 10  |-  ( w D z  ->  ( 1st `  z )  C_  ( 1st `  w ) )
9385simprbi 462 . . . . . . . . . 10  |-  ( v D w  ->  ( 1st `  w )  C_  ( 1st `  v ) )
9492, 93sylan9ssr 3503 . . . . . . . . 9  |-  ( ( v D w  /\  w D z )  -> 
( 1st `  z
)  C_  ( 1st `  v ) )
952, 3, 63, 71filnetlem1 30436 . . . . . . . . 9  |-  ( v D z  <->  ( (
v  e.  H  /\  z  e.  H )  /\  ( 1st `  z
)  C_  ( 1st `  v ) ) )
9691, 94, 95sylanbrc 662 . . . . . . . 8  |-  ( ( v D w  /\  w D z )  -> 
v D z )
9796ax-gen 1623 . . . . . . 7  |-  A. z
( ( v D w  /\  w D z )  ->  v D z )
9897gen2 1624 . . . . . 6  |-  A. v A. w A. z ( ( v D w  /\  w D z )  ->  v D
z )
99 cotr 5367 . . . . . 6  |-  ( ( D  o.  D ) 
C_  D  <->  A. v A. w A. z ( ( v D w  /\  w D z )  ->  v D
z ) )
10098, 99mpbir 209 . . . . 5  |-  ( D  o.  D )  C_  D
10183, 100jctil 535 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( ( D  o.  D )  C_  D  /\  ( H  X.  H )  C_  ( `' D  o.  D
) ) )
102 filtop 20522 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
103 xpexg 6575 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  X  e.  F )  ->  ( F  X.  X )  e. 
_V )
104102, 103mpdan 666 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( F  X.  X )  e.  _V )
105104, 30ssexd 4584 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  H  e.  _V )
106 xpexg 6575 . . . . . . 7  |-  ( ( H  e.  _V  /\  H  e.  _V )  ->  ( H  X.  H
)  e.  _V )
107105, 105, 106syl2anc 659 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( H  X.  H )  e.  _V )
108 ssexg 4583 . . . . . 6  |-  ( ( D  C_  ( H  X.  H )  /\  ( H  X.  H )  e. 
_V )  ->  D  e.  _V )
10913, 107, 108sylancr 661 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  D  e.  _V )
11021isdir 16061 . . . . 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
) ) ) ) )
111109, 110syl 16 . . . 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 )
) ) ) )
11233, 101, 111mpbir2and 920 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  D  e.  DirRel )
11330, 112jca 530 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) )
11421, 113pm3.2i 453 1  |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X
)  /\  D  e.  DirRel ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   <.cop 4022   U.cuni 4235   U_ciun 4315   class class class wbr 4439   {copab 4496    _I cid 4779    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990    o. ccom 4992   Rel wrel 4993   ` cfv 5570   1stc1st 6771   DirRelcdir 16057   Filcfil 20512
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-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-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-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  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-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  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-1st 6773  df-dir 16059  df-fbas 18611  df-fil 20513
This theorem is referenced by:  filnetlem4  30439
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