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

Theorem ustuqtop1 19816
Description: Lemma for ustuqtop 19821, similar to ssnei2 18720 (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 1039 . . . . . 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 1209 . . . . 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 754 . . . . 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 19779 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
52, 3, 4syl2anc 661 . . . . . 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 1040 . . . . . . . . 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 1209 . . . . . . . 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 4018 . . . . . . 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 993 . . . . . . . 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 1209 . . . . . . 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 4945 . . . . . . 7  |-  ( ( { p }  C_  X  /\  b  C_  X
)  ->  ( {
p }  X.  b
)  C_  ( X  X.  X ) )
128, 10, 11syl2anc 661 . . . . . 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 3532 . . . . 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 3519 . . . . . 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 19780 . . . . . 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 429 . . . . 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 1221 . . . 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 992 . . . . . 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 1209 . . . . 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 3526 . . . . . . 7  |-  ( a 
C_  b  <->  ( a  u.  b )  =  b )
2221biimpi 194 . . . . . 6  |-  ( a 
C_  b  ->  (
a  u.  b )  =  b )
23 id 22 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  a  =  ( w " { p } ) )
24 inidm 3559 . . . . . . . . . . 11  |-  ( { p }  i^i  {
p } )  =  { p }
25 vex 2975 . . . . . . . . . . . 12  |-  p  e. 
_V
2625snnz 3993 . . . . . . . . . . 11  |-  { p }  =/=  (/)
2724, 26eqnetri 2625 . . . . . . . . . 10  |-  ( { p }  i^i  {
p } )  =/=  (/)
28 xpima2 5282 . . . . . . . . . 10  |-  ( ( { p }  i^i  { p } )  =/=  (/)  ->  ( ( { p }  X.  b
) " { p } )  =  b )
2927, 28mp1i 12 . . . . . . . . 9  |-  ( a  =  ( w " { p } )  ->  ( ( { p }  X.  b
) " { p } )  =  b )
3029eqcomd 2448 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  b  =  ( ( { p }  X.  b ) " {
p } ) )
3123, 30uneq12d 3511 . . . . . . 7  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w " {
p } )  u.  ( ( { p }  X.  b ) " { p } ) ) )
32 imaundir 5250 . . . . . . 7  |-  ( ( w  u.  ( { p }  X.  b
) ) " {
p } )  =  ( ( w " { p } )  u.  ( ( { p }  X.  b
) " { p } ) )
3331, 32syl6eqr 2493 . . . . . 6  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3422, 33sylan9req 2496 . . . . 5  |-  ( ( a  C_  b  /\  a  =  ( w " { p } ) )  ->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3520, 34sylancom 667 . . . 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 5164 . . . . . 6  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( u " {
p } )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3736eqeq2d 2454 . . . . 5  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( b  =  ( u " { p } )  <->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) ) )
3837rspcev 3073 . . . 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 661 . . 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 2975 . . . . . 6  |-  a  e. 
_V
41 utopustuq.1 . . . . . . 7  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
4241ustuqtoplem 19814 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4340, 42mpan2 671 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4443biimpa 484 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
45443ad2antl1 1150 . . 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 2862 . 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 2975 . . . . 5  |-  b  e. 
_V
4841ustuqtoplem 19814 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
4947, 48mpan2 671 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
50493ad2ant1 1009 . . 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 465 . 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 232 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   _Vcvv 2972    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   {csn 3877    e. cmpt 4350    X. cxp 4838   ran crn 4841   "cima 4843   ` cfv 5418  UnifOncust 19774
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ust 19775
This theorem is referenced by:  ustuqtop4  19819  ustuqtop  19821  utopsnneiplem  19822
  Copyright terms: Public domain W3C validator