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Theorem ustuqtop0 20471
Description: Lemma for ustuqtop 20477 (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 20459 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U  /\  p  e.  X )  ->  (
v " { p } )  C_  X
)
213expa 1191 . . . . . . 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 3109 . . . . . . 7  |-  v  e. 
_V
5 imaexg 6711 . . . . . . 7  |-  ( v  e.  _V  ->  (
v " { p } )  e.  _V )
6 elpwg 4011 . . . . . . 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 2871 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  A. v  e.  U  ( v " { p } )  e.  ~P X )
10 eqid 2460 . . . . 5  |-  ( v  e.  U  |->  ( v
" { p }
) )  =  ( v  e.  U  |->  ( v " { p } ) )
1110rnmptss 6041 . . . 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 6121 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
14 rnexg 6706 . . . . 5  |-  ( ( v  e.  U  |->  ( v " { p } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
15 elpwg 4011 . . . . 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 6036 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 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    C_ wss 3469   ~Pcpw 4003   {csn 4020    |-> cmpt 4498   ran crn 4993   "cima 4995   -->wf 5575   ` cfv 5579  UnifOncust 20430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ust 20431
This theorem is referenced by:  ustuqtop  20477  utopsnneiplem  20478
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