MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtop5 Structured version   Unicode version

Theorem ustuqtop5 18267
Description: Lemma for ustuqtop 18268 (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 18228 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
21adantr 452 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  X.  X )  e.  U )
3 snssi 3934 . . . . . . . . 9  |-  ( p  e.  X  ->  { p }  C_  X )
4 dfss 3327 . . . . . . . . 9  |-  ( { p }  C_  X  <->  { p }  =  ( { p }  i^i  X ) )
53, 4sylib 189 . . . . . . . 8  |-  ( p  e.  X  ->  { p }  =  ( {
p }  i^i  X
) )
6 incom 3525 . . . . . . . 8  |-  ( { p }  i^i  X
)  =  ( X  i^i  { p }
)
75, 6syl6req 2484 . . . . . . 7  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =  { p } )
8 snnzg 3913 . . . . . . 7  |-  ( p  e.  X  ->  { p }  =/=  (/) )
97, 8eqnetrd 2616 . . . . . 6  |-  ( p  e.  X  ->  ( X  i^i  { p }
)  =/=  (/) )
109adantl 453 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  i^i  { p }
)  =/=  (/) )
11 xpima2 5307 . . . . 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 2440 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  =  ( ( X  X.  X ) " { p } ) )
14 imaeq1 5190 . . . . 5  |-  ( w  =  ( X  X.  X )  ->  (
w " { p } )  =  ( ( X  X.  X
) " { p } ) )
1514eqeq2d 2446 . . . 4  |-  ( w  =  ( X  X.  X )  ->  ( X  =  ( w " { p } )  <-> 
X  =  ( ( X  X.  X )
" { p }
) ) )
1615rspcev 3044 . . 3  |-  ( ( ( X  X.  X
)  e.  U  /\  X  =  ( ( X  X.  X ) " { p } ) )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
172, 13, 16syl2anc 643 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  E. w  e.  U  X  =  ( w " {
p } ) )
18 elfvex 5750 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
1918adantr 452 . . 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 18261 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  X  e.  _V )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2219, 21mpdan 650 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( X  e.  ( N `  p )  <->  E. w  e.  U  X  =  ( w " {
p } ) ) )
2317, 22mpbird 224 1  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806    e. cmpt 4258    X. cxp 4868   ran crn 4871   "cima 4873   ` cfv 5446  UnifOncust 18221
This theorem is referenced by:  ustuqtop  18268  utopsnneiplem  18269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ust 18222
  Copyright terms: Public domain W3C validator