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

Theorem ustuqtop3 21313
Description: Lemma for ustuqtop 21316, similar to elnei 20182. (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 5719 . . . . . . 7  |-  (  _I  |`  X )  Fn  X
2 fnsnfv 5953 . . . . . . 7  |-  ( ( (  _I  |`  X )  Fn  X  /\  p  e.  X )  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
31, 2mpan 681 . . . . . 6  |-  ( p  e.  X  ->  { ( (  _I  |`  X ) `
 p ) }  =  ( (  _I  |`  X ) " {
p } ) )
43ad4antlr 744 . . . . 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 781 . . . . . . 7  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  U  e.  (UnifOn `  X ) )
6 simplr 767 . . . . . . 7  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  w  e.  U )
7 ustdiag 21278 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  (  _I  |`  X )  C_  w )
85, 6, 7syl2anc 671 . . . . . 6  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  (  _I  |`  X )  C_  w
)
9 imass1 5225 . . . . . 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 3478 . . . 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 5902 . . . . 5  |-  ( (  _I  |`  X ) `  p )  e.  _V
1312snss 4109 . . . 4  |-  ( ( (  _I  |`  X ) `
 p )  e.  ( w " {
p } )  <->  { (
(  _I  |`  X ) `
 p ) } 
C_  ( w " { p } ) )
1411, 13sylibr 217 . . 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 6119 . . . . 5  |-  ( p  e.  X  ->  (
(  _I  |`  X ) `
 p )  =  p )
1615eqcomd 2468 . . . 4  |-  ( p  e.  X  ->  p  =  ( (  _I  |`  X ) `  p
) )
1716ad4antlr 744 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  =  ( (  _I  |`  X ) `
 p ) )
18 simpr 467 . . 3  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  a  =  ( w " {
p } ) )
1914, 17, 183eltr4d 2555 . 2  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  e.  a )
20 vex 3060 . . . 4  |-  a  e. 
_V
21 utopustuq.1 . . . . 5  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2221ustuqtoplem 21309 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2320, 22mpan2 682 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
2423biimpa 491 . 2  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
2519, 24r19.29a 2944 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750   _Vcvv 3057    C_ wss 3416   {csn 3980    |-> cmpt 4477    _I cid 4766   ran crn 4857    |` cres 4858   "cima 4859    Fn wfn 5600   ` cfv 5605  UnifOncust 21269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ust 21270
This theorem is referenced by:  ustuqtop  21316  utopsnneiplem  21317
  Copyright terms: Public domain W3C validator