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Theorem ustuqtop0 20912
Description: Lemma for ustuqtop 20918 (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 20900 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U  /\  p  e.  X )  ->  (
v " { p } )  C_  X
)
213expa 1194 . . . . . . 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 6710 . . . . . . 7  |-  ( v  e.  _V  ->  (
v " { p } )  e.  _V )
6 elpwg 4007 . . . . . . 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 2868 . . . 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 6036 . . . 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 6117 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
14 rnexg 6705 . . . . 5  |-  ( ( v  e.  U  |->  ( v " { p } ) )  e. 
_V  ->  ran  ( v  e.  U  |->  ( v
" { p }
) )  e.  _V )
15 elpwg 4007 . . . . 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 463 . . 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 6031 1  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   {csn 4016    |-> cmpt 4497   ran crn 4989   "cima 4991   -->wf 5566   ` cfv 5570  UnifOncust 20871
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-rep 4550  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-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-ust 20872
This theorem is referenced by:  ustuqtop  20918  utopsnneiplem  20919
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