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Theorem ustincl 20442
Description: A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
Assertion
Ref Expression
ustincl  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )

Proof of Theorem ustincl
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5891 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 20438 . . . . . . . 8  |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X
)  <->  ( U  C_  ~P ( X  X.  X
)  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) ) )
31, 2syl 16 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  e.  (UnifOn `  X )  <->  ( U  C_  ~P ( X  X.  X )  /\  ( X  X.  X
)  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) ) )
43ibi 241 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  C_ 
~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) )
54simp3d 1010 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) )
6 sseq1 3525 . . . . . . . . 9  |-  ( v  =  V  ->  (
v  C_  w  <->  V  C_  w
) )
76imbi1d 317 . . . . . . . 8  |-  ( v  =  V  ->  (
( v  C_  w  ->  w  e.  U )  <-> 
( V  C_  w  ->  w  e.  U ) ) )
87ralbidv 2903 . . . . . . 7  |-  ( v  =  V  ->  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  <->  A. w  e.  ~P  ( X  X.  X
) ( V  C_  w  ->  w  e.  U
) ) )
9 ineq1 3693 . . . . . . . . 9  |-  ( v  =  V  ->  (
v  i^i  w )  =  ( V  i^i  w ) )
109eleq1d 2536 . . . . . . . 8  |-  ( v  =  V  ->  (
( v  i^i  w
)  e.  U  <->  ( V  i^i  w )  e.  U
) )
1110ralbidv 2903 . . . . . . 7  |-  ( v  =  V  ->  ( A. w  e.  U  ( v  i^i  w
)  e.  U  <->  A. w  e.  U  ( V  i^i  w )  e.  U
) )
12 sseq2 3526 . . . . . . . 8  |-  ( v  =  V  ->  (
(  _I  |`  X ) 
C_  v  <->  (  _I  |`  X )  C_  V
) )
13 cnveq 5174 . . . . . . . . 9  |-  ( v  =  V  ->  `' v  =  `' V
)
1413eleq1d 2536 . . . . . . . 8  |-  ( v  =  V  ->  ( `' v  e.  U  <->  `' V  e.  U ) )
15 sseq2 3526 . . . . . . . . 9  |-  ( v  =  V  ->  (
( w  o.  w
)  C_  v  <->  ( w  o.  w )  C_  V
) )
1615rexbidv 2973 . . . . . . . 8  |-  ( v  =  V  ->  ( E. w  e.  U  ( w  o.  w
)  C_  v  <->  E. w  e.  U  ( w  o.  w )  C_  V
) )
1712, 14, 163anbi123d 1299 . . . . . . 7  |-  ( v  =  V  ->  (
( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v )  <->  ( (  _I  |`  X )  C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w )  C_  V
) ) )
188, 11, 173anbi123d 1299 . . . . . 6  |-  ( v  =  V  ->  (
( A. w  e. 
~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U )  /\  A. w  e.  U  (
v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) )  <-> 
( A. w  e. 
~P  ( X  X.  X ) ( V 
C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( V  i^i  w )  e.  U  /\  ( (  _I  |`  X )  C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w )  C_  V
) ) ) )
1918rspcv 3210 . . . . 5  |-  ( V  e.  U  ->  ( A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) )  ->  ( A. w  e.  ~P  ( X  X.  X ) ( V 
C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( V  i^i  w )  e.  U  /\  ( (  _I  |`  X )  C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w )  C_  V
) ) ) )
205, 19mpan9 469 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( A. w  e.  ~P  ( X  X.  X
) ( V  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( V  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  V ) ) )
2120simp2d 1009 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  A. w  e.  U  ( V  i^i  w )  e.  U
)
22 ineq2 3694 . . . . 5  |-  ( w  =  W  ->  ( V  i^i  w )  =  ( V  i^i  W
) )
2322eleq1d 2536 . . . 4  |-  ( w  =  W  ->  (
( V  i^i  w
)  e.  U  <->  ( V  i^i  W )  e.  U
) )
2423rspcv 3210 . . 3  |-  ( W  e.  U  ->  ( A. w  e.  U  ( V  i^i  w
)  e.  U  -> 
( V  i^i  W
)  e.  U ) )
2521, 24mpan9 469 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )
26253impa 1191 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010    _I cid 4790    X. cxp 4997   `'ccnv 4998    |` cres 5001    o. ccom 5003   ` cfv 5586  UnifOncust 20434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594  df-ust 20435
This theorem is referenced by:  ustexsym  20450  trust  20464  utoptop  20469  restutopopn  20473  ustuqtop2  20477
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