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Theorem ustuqtop3 20474
Description: Lemma for ustuqtop 20477, similar to elnei 19371 (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
ustuqtop3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
Distinct variable groups:    v, p, U    X, p, v, a    N, a, p    v, a, U    X, a
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fnresi 5689 . . . . . . 7  |-  (  _I  |`  X )  Fn  X
2 fnsnfv 5918 . . . . . . 7  |-  ( ( (  _I  |`  X )  Fn  X  /\  p  e.  X )  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
31, 2mpan 670 . . . . . 6  |-  ( p  e.  X  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
43ad4antlr 732 . . . . 5  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  { (
(  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
5 simp-4l 765 . . . . . . 7  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  U  e.  (UnifOn `  X ) )
6 simplr 754 . . . . . . 7  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  w  e.  U )
7 ustdiag 20439 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  (  _I  |`  X )  C_  w )
85, 6, 7syl2anc 661 . . . . . 6  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  (  _I  |`  X )  C_  w
)
9 imass1 5362 . . . . . 6  |-  ( (  _I  |`  X )  C_  w  ->  ( (  _I  |`  X ) " { p } ) 
C_  ( w " { p } ) )
108, 9syl 16 . . . . 5  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( (  _I  |`  X ) " { p } ) 
C_  ( w " { p } ) )
114, 10eqsstrd 3531 . . . 4  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  { (
(  _I  |`  X ) `
 p ) } 
C_  ( w " { p } ) )
12 fvex 5867 . . . . 5  |-  ( (  _I  |`  X ) `  p )  e.  _V
1312snss 4144 . . . 4  |-  ( ( (  _I  |`  X ) `
 p )  e.  ( w " {
p } )  <->  { (
(  _I  |`  X ) `
 p ) } 
C_  ( w " { p } ) )
1411, 13sylibr 212 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( (  _I  |`  X ) `  p )  e.  ( w " { p } ) )
15 fvresi 6078 . . . . 5  |-  ( p  e.  X  ->  (
(  _I  |`  X ) `
 p )  =  p )
1615eqcomd 2468 . . . 4  |-  ( p  e.  X  ->  p  =  ( (  _I  |`  X ) `  p
) )
1716ad4antlr 732 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  =  ( (  _I  |`  X ) `
 p ) )
18 simpr 461 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  a  =  ( w " {
p } ) )
1914, 17, 183eltr4d 2563 . 2  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  e.  a )
20 vex 3109 . . . 4  |-  a  e. 
_V
21 utopustuq.1 . . . . 5  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2221ustuqtoplem 20470 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2320, 22mpan2 671 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2423biimpa 484 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
2519, 24r19.29a 2996 1  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106    C_ wss 3469   {csn 4020    |-> cmpt 4498    _I cid 4783   ran crn 4993    |` cres 4994   "cima 4995    Fn wfn 5574   ` cfv 5579  UnifOncust 20430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ust 20431
This theorem is referenced by:  ustuqtop  20477  utopsnneiplem  20478
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