MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtop3 Structured version   Unicode version

Theorem ustuqtop3 19816
Description: Lemma for ustuqtop 19819, similar to elnei 18713 (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 5526 . . . . . . 7  |-  (  _I  |`  X )  Fn  X
2 fnsnfv 5749 . . . . . . 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 19781 . . . . . . 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 5201 . . . . . 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 3388 . . . 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 5699 . . . . 5  |-  ( (  _I  |`  X ) `  p )  e.  _V
1312snss 3997 . . . 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 5902 . . . . 5  |-  ( p  e.  X  ->  (
(  _I  |`  X ) `
 p )  =  p )
1615eqcomd 2446 . . . 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 2522 . 2  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  e.  a )
20 vex 2973 . . . 4  |-  a  e. 
_V
21 utopustuq.1 . . . . 5  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2221ustuqtoplem 19812 . . . 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 2860 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 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970    C_ wss 3326   {csn 3875    e. cmpt 4348    _I cid 4629   ran crn 4839    |` cres 4840   "cima 4841    Fn wfn 5411   ` cfv 5416  UnifOncust 19772
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ust 19773
This theorem is referenced by:  ustuqtop  19819  utopsnneiplem  19820
  Copyright terms: Public domain W3C validator