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Theorem ustuqtop5 20575
Description: Lemma for ustuqtop 20576 (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 20536 . . . 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 4171 . . . . . . . . 9  |-  ( p  e.  X  ->  { p }  C_  X )
4 dfss 3491 . . . . . . . . 9  |-  ( { p }  C_  X  <->  { p }  =  ( { p }  i^i  X ) )
53, 4sylib 196 . . . . . . . 8  |-  ( p  e.  X  ->  { p }  =  ( {
p }  i^i  X
) )
6 incom 3691 . . . . . . . 8  |-  ( { p }  i^i  X
)  =  ( X  i^i  { p }
)
75, 6syl6req 2525 . . . . . . 7  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =  { p } )
8 snnzg 4144 . . . . . . 7  |-  ( p  e.  X  ->  { p }  =/=  (/) )
97, 8eqnetrd 2760 . . . . . 6  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =/=  (/) )
109adantl 466 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  i^i  { p }
)  =/=  (/) )
11 xpima2 5451 . . . . 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 2475 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  =  ( ( X  X.  X ) " { p } ) )
14 imaeq1 5332 . . . . 5  |-  ( w  =  ( X  X.  X )  ->  (
w " { p } )  =  ( ( X  X.  X
) " { p } ) )
1514eqeq2d 2481 . . . 4  |-  ( w  =  ( X  X.  X )  ->  ( X  =  ( w " { p } )  <-> 
X  =  ( ( X  X.  X )
" { p }
) ) )
1615rspcev 3214 . . 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 5893 . . . 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 20569 . . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5588  UnifOncust 20529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ust 20530
This theorem is referenced by:  ustuqtop  20576  utopsnneiplem  20577
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