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Theorem ustuqtop3 21038
Description: Lemma for ustuqtop 21041, similar to elnei 19905 (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 5679 . . . . . . 7  |-  (  _I  |`  X )  Fn  X
2 fnsnfv 5909 . . . . . . 7  |-  ( ( (  _I  |`  X )  Fn  X  /\  p  e.  X )  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
31, 2mpan 668 . . . . . 6  |-  ( p  e.  X  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
43ad4antlr 731 . . . . 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 768 . . . . . . 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 21003 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  (  _I  |`  X )  C_  w )
85, 6, 7syl2anc 659 . . . . . 6  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  (  _I  |`  X )  C_  w
)
9 imass1 5191 . . . . . 6  |-  ( (  _I  |`  X )  C_  w  ->  ( (  _I  |`  X ) " { p } ) 
C_  ( w " { p } ) )
108, 9syl 17 . . . . 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 3476 . . . 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 5859 . . . . 5  |-  ( (  _I  |`  X ) `  p )  e.  _V
1312snss 4096 . . . 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 6077 . . . . 5  |-  ( p  e.  X  ->  (
(  _I  |`  X ) `
 p )  =  p )
1615eqcomd 2410 . . . 4  |-  ( p  e.  X  ->  p  =  ( (  _I  |`  X ) `  p
) )
1716ad4antlr 731 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  =  ( (  _I  |`  X ) `
 p ) )
18 simpr 459 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  a  =  ( w " {
p } ) )
1914, 17, 183eltr4d 2505 . 2  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  e.  a )
20 vex 3062 . . . 4  |-  a  e. 
_V
21 utopustuq.1 . . . . 5  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2221ustuqtoplem 21034 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2320, 22mpan2 669 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2423biimpa 482 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
2519, 24r19.29a 2949 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 367    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059    C_ wss 3414   {csn 3972    |-> cmpt 4453    _I cid 4733   ran crn 4824    |` cres 4825   "cima 4826    Fn wfn 5564   ` cfv 5569  UnifOncust 20994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ust 20995
This theorem is referenced by:  ustuqtop  21041  utopsnneiplem  21042
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