MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtop0 Structured version   Unicode version

Theorem ustuqtop0 19948
Description: Lemma for ustuqtop 19954 (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
Assertion
Ref Expression
ustuqtop0  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
Distinct variable groups:    v, p, U    X, p, v    N, p
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop0
StepHypRef Expression
1 ustimasn 19936 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U  /\  p  e.  X )  ->  (
v " { p } )  C_  X
)
213expa 1188 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  /\  p  e.  X )  ->  (
v " { p } )  C_  X
)
32an32s 802 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  v  e.  U )  ->  (
v " { p } )  C_  X
)
4 vex 3081 . . . . . . 7  |-  v  e. 
_V
5 imaexg 6626 . . . . . . 7  |-  ( v  e.  _V  ->  (
v " { p } )  e.  _V )
6 elpwg 3977 . . . . . . 7  |-  ( ( v " { p } )  e.  _V  ->  ( ( v " { p } )  e.  ~P X  <->  ( v " { p } ) 
C_  X ) )
74, 5, 6mp2b 10 . . . . . 6  |-  ( ( v " { p } )  e.  ~P X 
<->  ( v " {
p } )  C_  X )
83, 7sylibr 212 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  v  e.  U )  ->  (
v " { p } )  e.  ~P X )
98ralrimiva 2830 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  A. v  e.  U  ( v " { p } )  e.  ~P X )
10 eqid 2454 . . . . 5  |-  ( v  e.  U  |->  ( v
" { p }
) )  =  ( v  e.  U  |->  ( v " { p } ) )
1110rnmptss 5982 . . . 4  |-  ( A. v  e.  U  (
v " { p } )  e.  ~P X  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  C_  ~P X )
129, 11syl 16 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ran  ( v  e.  U  |->  ( v " {
p } ) ) 
C_  ~P X )
13 mptexg 6057 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
14 rnexg 6621 . . . . 5  |-  ( ( v  e.  U  |->  ( v " { p } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
15 elpwg 3977 . . . . 5  |-  ( ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  _V  ->  ( ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  ~P ~P X  <->  ran  ( v  e.  U  |->  ( v " {
p } ) ) 
C_  ~P X ) )
1613, 14, 153syl 20 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  ~P ~P X  <->  ran  ( v  e.  U  |->  ( v " {
p } ) ) 
C_  ~P X ) )
1716adantr 465 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  ~P ~P X  <->  ran  ( v  e.  U  |->  ( v " {
p } ) ) 
C_  ~P X ) )
1812, 17mpbird 232 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ran  ( v  e.  U  |->  ( v " {
p } ) )  e.  ~P ~P X
)
19 utopustuq.1 . 2  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2018, 19fmptd 5977 1  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    C_ wss 3437   ~Pcpw 3969   {csn 3986    |-> cmpt 4459   ran crn 4950   "cima 4952   -->wf 5523   ` cfv 5527  UnifOncust 19907
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-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-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  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-ust 19908
This theorem is referenced by:  ustuqtop  19954  utopsnneiplem  19955
  Copyright terms: Public domain W3C validator