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Theorem ustuqtop1 21256
Description: Lemma for ustuqtop 21261, similar to ssnei2 20132. (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 1059 . . . . . 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 1232 . . . . 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 762 . . . . 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 21219 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
52, 3, 4syl2anc 667 . . . . . 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 1060 . . . . . . . . 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 1232 . . . . . . . 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 4117 . . . . . . 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 1013 . . . . . . . 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 1232 . . . . . . 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 4940 . . . . . . 7  |-  ( ( { p }  C_  X  /\  b  C_  X
)  ->  ( {
p }  X.  b
)  C_  ( X  X.  X ) )
128, 10, 11syl2anc 667 . . . . . 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 3610 . . . . 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 3597 . . . . . 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 21220 . . . . . 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 431 . . . . 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 1271 . . . 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 1012 . . . . . 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 1232 . . . . 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 3604 . . . . . . 7  |-  ( a 
C_  b  <->  ( a  u.  b )  =  b )
2221biimpi 198 . . . . . 6  |-  ( a 
C_  b  ->  (
a  u.  b )  =  b )
23 id 22 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  a  =  ( w " { p } ) )
24 inidm 3641 . . . . . . . . . . 11  |-  ( { p }  i^i  {
p } )  =  { p }
25 vex 3048 . . . . . . . . . . . 12  |-  p  e. 
_V
2625snnz 4090 . . . . . . . . . . 11  |-  { p }  =/=  (/)
2724, 26eqnetri 2694 . . . . . . . . . 10  |-  ( { p }  i^i  {
p } )  =/=  (/)
28 xpima2 5281 . . . . . . . . . 10  |-  ( ( { p }  i^i  { p } )  =/=  (/)  ->  ( ( { p }  X.  b
) " { p } )  =  b )
2927, 28mp1i 13 . . . . . . . . 9  |-  ( a  =  ( w " { p } )  ->  ( ( { p }  X.  b
) " { p } )  =  b )
3029eqcomd 2457 . . . . . . . 8  |-  ( a  =  ( w " { p } )  ->  b  =  ( ( { p }  X.  b ) " {
p } ) )
3123, 30uneq12d 3589 . . . . . . 7  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w " {
p } )  u.  ( ( { p }  X.  b ) " { p } ) ) )
32 imaundir 5249 . . . . . . 7  |-  ( ( w  u.  ( { p }  X.  b
) ) " {
p } )  =  ( ( w " { p } )  u.  ( ( { p }  X.  b
) " { p } ) )
3331, 32syl6eqr 2503 . . . . . 6  |-  ( a  =  ( w " { p } )  ->  ( a  u.  b )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3422, 33sylan9req 2506 . . . . 5  |-  ( ( a  C_  b  /\  a  =  ( w " { p } ) )  ->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3520, 34sylancom 673 . . . 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 5163 . . . . . 6  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( u " {
p } )  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) )
3736eqeq2d 2461 . . . . 5  |-  ( u  =  ( w  u.  ( { p }  X.  b ) )  -> 
( b  =  ( u " { p } )  <->  b  =  ( ( w  u.  ( { p }  X.  b ) ) " { p } ) ) )
3837rspcev 3150 . . . 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 667 . . 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 3048 . . . . . 6  |-  a  e. 
_V
41 utopustuq.1 . . . . . . 7  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
4241ustuqtoplem 21254 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4340, 42mpan2 677 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
4443biimpa 487 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
45443ad2antl1 1170 . . 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 2932 . 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 3048 . . . . 5  |-  b  e. 
_V
4841ustuqtoplem 21254 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
4947, 48mpan2 677 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
50493ad2ant1 1029 . . 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 467 . 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 236 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045    u. cun 3402    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968    |-> cmpt 4461    X. cxp 4832   ran crn 4835   "cima 4837   ` cfv 5582  UnifOncust 21214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ust 21215
This theorem is referenced by:  ustuqtop4  21259  ustuqtop  21261  utopsnneiplem  21262
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