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Theorem ustuqtop5 19832
Description: Lemma for ustuqtop 19833 (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 19793 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
21adantr 465 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  X.  X )  e.  U )
3 snssi 4029 . . . . . . . . 9  |-  ( p  e.  X  ->  { p }  C_  X )
4 dfss 3355 . . . . . . . . 9  |-  ( { p }  C_  X  <->  { p }  =  ( { p }  i^i  X ) )
53, 4sylib 196 . . . . . . . 8  |-  ( p  e.  X  ->  { p }  =  ( {
p }  i^i  X
) )
6 incom 3555 . . . . . . . 8  |-  ( { p }  i^i  X
)  =  ( X  i^i  { p }
)
75, 6syl6req 2492 . . . . . . 7  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =  { p } )
8 snnzg 4004 . . . . . . 7  |-  ( p  e.  X  ->  { p }  =/=  (/) )
97, 8eqnetrd 2638 . . . . . 6  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =/=  (/) )
109adantl 466 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  i^i  { p }
)  =/=  (/) )
11 xpima2 5294 . . . . 5  |-  ( ( X  i^i  { p } )  =/=  (/)  ->  (
( X  X.  X
) " { p } )  =  X )
1210, 11syl 16 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
( X  X.  X
) " { p } )  =  X )
1312eqcomd 2448 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  =  ( ( X  X.  X ) " { p } ) )
14 imaeq1 5176 . . . . 5  |-  ( w  =  ( X  X.  X )  ->  (
w " { p } )  =  ( ( X  X.  X
) " { p } ) )
1514eqeq2d 2454 . . . 4  |-  ( w  =  ( X  X.  X )  ->  ( X  =  ( w " { p } )  <-> 
X  =  ( ( X  X.  X )
" { p }
) ) )
1615rspcev 3085 . . 3  |-  ( ( ( X  X.  X
)  e.  U  /\  X  =  ( ( X  X.  X ) " { p } ) )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
172, 13, 16syl2anc 661 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
18 elfvex 5729 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
1918adantr 465 . . 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 19826 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  X  e.  _V )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2219, 21mpdan 668 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2317, 22mpbird 232 1  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   _Vcvv 2984    i^i cin 3339    C_ wss 3340   (/)c0 3649   {csn 3889    e. cmpt 4362    X. cxp 4850   ran crn 4853   "cima 4855   ` cfv 5430  UnifOncust 19786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ust 19787
This theorem is referenced by:  ustuqtop  19833  utopsnneiplem  19834
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