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Theorem ustuqtop1 20495
Description: Lemma for ustuqtop 20500, similar to ssnei2 19399 (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 1047 . . . . . 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 1218 . . . . 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 20458 . . . . . . 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 1048 . . . . . . . . 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 1218 . . . . . . . 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 4172 . . . . . . 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 1001 . . . . . . . 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 1218 . . . . . . 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 5107 . . . . . . 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 3680 . . . . 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 3667 . . . . . 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 20459 . . . . . 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 1231 . . . 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 1000 . . . . . 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 1218 . . . . 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 3674 . . . . . . 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 3707 . . . . . . . . . . 11  |-  ( { p }  i^i  {
p } )  =  { p }
25 vex 3116 . . . . . . . . . . . 12  |-  p  e. 
_V
2625snnz 4145 . . . . . . . . . . 11  |-  { p }  =/=  (/)
2724, 26eqnetri 2763 . . . . . . . . . 10  |-  ( { p }  i^i  {
p } )  =/=  (/)
28 xpima2 5450 . . . . . . . . . 10  |-  ( ( { p }  i^i  { p } )  =/=  (/)  ->  ( ( { p }  X.  b
) " { p } )  =  b )
2927, 28mp1i 12 . . . . . . . . 9  |-  ( a  =  ( w " { p } )  ->  ( ( { p }  X.  b
) " { p } )  =  b )
3029eqcomd 2475 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  b  =  ( ( { p }  X.  b ) " {
p } ) )
3123, 30uneq12d 3659 . . . . . . 7  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w " {
p } )  u.  ( ( { p }  X.  b ) " { p } ) ) )
32 imaundir 5418 . . . . . . 7  |-  ( ( w  u.  ( { p }  X.  b
) ) " {
p } )  =  ( ( w " { p } )  u.  ( ( { p }  X.  b
) " { p } ) )
3331, 32syl6eqr 2526 . . . . . 6  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3422, 33sylan9req 2529 . . . . 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 5331 . . . . . 6  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( u " {
p } )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3736eqeq2d 2481 . . . . 5  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( b  =  ( u " { p } )  <->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) ) )
3837rspcev 3214 . . . 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 3116 . . . . . 6  |-  a  e. 
_V
41 utopustuq.1 . . . . . . 7  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
4241ustuqtoplem 20493 . . . . . 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 1158 . . 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 3003 . 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 3116 . . . . 5  |-  b  e. 
_V
4841ustuqtoplem 20493 . . . . 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 1017 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5587  UnifOncust 20453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ust 20454
This theorem is referenced by:  ustuqtop4  20498  ustuqtop  20500  utopsnneiplem  20501
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