MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtop1 Structured version   Unicode version

Theorem ustuqtop1 21187
Description: Lemma for ustuqtop 21192, similar to ssnei2 20063 (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
ustuqtop1  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
Distinct variable groups:    v, p, U    X, p, v    a,
b, p, N    v,
a, U, b    X, a, b
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop1
Dummy variables  w  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1l 1056 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  ( a  e.  ( N `  p
)  /\  w  e.  U  /\  a  =  ( w " { p } ) ) )  ->  U  e.  (UnifOn `  X ) )
213anassrs 1228 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  U  e.  (UnifOn `  X ) )
3 simplr 760 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  w  e.  U )
4 ustssxp 21150 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
52, 3, 4syl2anc 665 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  w  C_  ( X  X.  X ) )
6 simpl1r 1057 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  ( a  e.  ( N `  p
)  /\  w  e.  U  /\  a  =  ( w " { p } ) ) )  ->  p  e.  X
)
763anassrs 1228 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  p  e.  X )
87snssd 4148 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  { p }  C_  X )
9 simpl3 1010 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  ( a  e.  ( N `  p
)  /\  w  e.  U  /\  a  =  ( w " { p } ) ) )  ->  b  C_  X
)
1093anassrs 1228 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  b  C_  X )
11 xpss12 4960 . . . . . . 7  |-  ( ( { p }  C_  X  /\  b  C_  X
)  ->  ( {
p }  X.  b
)  C_  ( X  X.  X ) )
128, 10, 11syl2anc 665 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( {
p }  X.  b
)  C_  ( X  X.  X ) )
135, 12unssd 3648 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( w  u.  ( { p }  X.  b ) )  C_  ( X  X.  X
) )
14 ssun1 3635 . . . . . 6  |-  w  C_  ( w  u.  ( { p }  X.  b ) )
1514a1i 11 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  w  C_  (
w  u.  ( { p }  X.  b
) ) )
16 ustssel 21151 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  (
w  u.  ( { p }  X.  b
) )  C_  ( X  X.  X ) )  ->  ( w  C_  ( w  u.  ( { p }  X.  b ) )  -> 
( w  u.  ( { p }  X.  b ) )  e.  U ) )
1716imp 430 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  (
w  u.  ( { p }  X.  b
) )  C_  ( X  X.  X ) )  /\  w  C_  (
w  u.  ( { p }  X.  b
) ) )  -> 
( w  u.  ( { p }  X.  b ) )  e.  U )
182, 3, 13, 15, 17syl31anc 1267 . . . 4  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( w  u.  ( { p }  X.  b ) )  e.  U )
19 simpl2 1009 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  ( a  e.  ( N `  p
)  /\  w  e.  U  /\  a  =  ( w " { p } ) ) )  ->  a  C_  b
)
20193anassrs 1228 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  a  C_  b )
21 ssequn1 3642 . . . . . . 7  |-  ( a 
C_  b  <->  ( a  u.  b )  =  b )
2221biimpi 197 . . . . . 6  |-  ( a 
C_  b  ->  (
a  u.  b )  =  b )
23 id 23 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  a  =  ( w " { p } ) )
24 inidm 3677 . . . . . . . . . . 11  |-  ( { p }  i^i  {
p } )  =  { p }
25 vex 3090 . . . . . . . . . . . 12  |-  p  e. 
_V
2625snnz 4121 . . . . . . . . . . 11  |-  { p }  =/=  (/)
2724, 26eqnetri 2727 . . . . . . . . . 10  |-  ( { p }  i^i  {
p } )  =/=  (/)
28 xpima2 5301 . . . . . . . . . 10  |-  ( ( { p }  i^i  { p } )  =/=  (/)  ->  ( ( { p }  X.  b
) " { p } )  =  b )
2927, 28mp1i 13 . . . . . . . . 9  |-  ( a  =  ( w " { p } )  ->  ( ( { p }  X.  b
) " { p } )  =  b )
3029eqcomd 2437 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  b  =  ( ( { p }  X.  b ) " {
p } ) )
3123, 30uneq12d 3627 . . . . . . 7  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w " {
p } )  u.  ( ( { p }  X.  b ) " { p } ) ) )
32 imaundir 5269 . . . . . . 7  |-  ( ( w  u.  ( { p }  X.  b
) ) " {
p } )  =  ( ( w " { p } )  u.  ( ( { p }  X.  b
) " { p } ) )
3331, 32syl6eqr 2488 . . . . . 6  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3422, 33sylan9req 2491 . . . . 5  |-  ( ( a  C_  b  /\  a  =  ( w " { p } ) )  ->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3520, 34sylancom 671 . . . 4  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
36 imaeq1 5183 . . . . . 6  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( u " {
p } )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3736eqeq2d 2443 . . . . 5  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( b  =  ( u " { p } )  <->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) ) )
3837rspcev 3188 . . . 4  |-  ( ( ( w  u.  ( { p }  X.  b ) )  e.  U  /\  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3918, 35, 38syl2anc 665 . . 3  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
40 vex 3090 . . . . . 6  |-  a  e. 
_V
41 utopustuq.1 . . . . . . 7  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
4241ustuqtoplem 21185 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4340, 42mpan2 675 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4443biimpa 486 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
45443ad2antl1 1167 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
4639, 45r19.29a 2977 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
47 vex 3090 . . . . 5  |-  b  e. 
_V
4841ustuqtoplem 21185 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
4947, 48mpan2 675 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
50493ad2ant1 1026 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
5150adantr 466 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  ( b  e.  ( N `  p
)  <->  E. u  e.  U  b  =  ( u " { p } ) ) )
5246, 51mpbird 235 1  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   _Vcvv 3087    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002    |-> cmpt 4484    X. cxp 4852   ran crn 4855   "cima 4857   ` cfv 5601  UnifOncust 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ust 21146
This theorem is referenced by:  ustuqtop4  21190  ustuqtop  21192  utopsnneiplem  21193
  Copyright terms: Public domain W3C validator