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Theorem ustuqtop5 21184
Description: Lemma for ustuqtop 21185 (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
ustuqtop5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
Distinct variable groups:    v, p, U    X, p, v    N, p
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop5
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ustbasel 21145 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
21adantr 466 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  X.  X )  e.  U )
3 snssi 4138 . . . . . . . . 9  |-  ( p  e.  X  ->  { p }  C_  X )
4 dfss 3448 . . . . . . . . 9  |-  ( { p }  C_  X  <->  { p }  =  ( { p }  i^i  X ) )
53, 4sylib 199 . . . . . . . 8  |-  ( p  e.  X  ->  { p }  =  ( {
p }  i^i  X
) )
6 incom 3652 . . . . . . . 8  |-  ( { p }  i^i  X
)  =  ( X  i^i  { p }
)
75, 6syl6req 2478 . . . . . . 7  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =  { p } )
8 snnzg 4111 . . . . . . 7  |-  ( p  e.  X  ->  { p }  =/=  (/) )
97, 8eqnetrd 2715 . . . . . 6  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =/=  (/) )
109adantl 467 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  i^i  { p }
)  =/=  (/) )
11 xpima2 5292 . . . . 5  |-  ( ( X  i^i  { p } )  =/=  (/)  ->  (
( X  X.  X
) " { p } )  =  X )
1210, 11syl 17 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
( X  X.  X
) " { p } )  =  X )
1312eqcomd 2428 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  =  ( ( X  X.  X ) " { p } ) )
14 imaeq1 5174 . . . . 5  |-  ( w  =  ( X  X.  X )  ->  (
w " { p } )  =  ( ( X  X.  X
) " { p } ) )
1514eqeq2d 2434 . . . 4  |-  ( w  =  ( X  X.  X )  ->  ( X  =  ( w " { p } )  <-> 
X  =  ( ( X  X.  X )
" { p }
) ) )
1615rspcev 3179 . . 3  |-  ( ( ( X  X.  X
)  e.  U  /\  X  =  ( ( X  X.  X ) " { p } ) )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
172, 13, 16syl2anc 665 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
18 elfvex 5899 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
1918adantr 466 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  _V )
20 utopustuq.1 . . . 4  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 21178 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  X  e.  _V )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2219, 21mpdan 672 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2317, 22mpbird 235 1  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774   _Vcvv 3078    i^i cin 3432    C_ wss 3433   (/)c0 3758   {csn 3993    |-> cmpt 4475    X. cxp 4843   ran crn 4846   "cima 4848   ` cfv 5592  UnifOncust 21138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ust 21139
This theorem is referenced by:  ustuqtop  21185  utopsnneiplem  21186
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