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Theorem ustuqtop2 21305
Description: Lemma for ustuqtop 21309. (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
ustuqtop2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
Distinct variable groups:    v, p, U    X, p, v    N, p
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop2
Dummy variables  w  a  b  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-6l 785 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  ( U  e.  (UnifOn `  X )  /\  p  e.  X
) )
2 simp-7l 787 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  U  e.  (UnifOn `  X ) )
3 simp-4r 782 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  w  e.  U )
4 simplr 767 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  u  e.  U )
5 ustincl 21270 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  u  e.  U )  ->  (
w  i^i  u )  e.  U )
62, 3, 4, 5syl3anc 1276 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  ( w  i^i  u )  e.  U
)
7 simpllr 774 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  a  =  ( w " {
p } ) )
8 ineq12 3640 . . . . . . . . . . 11  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
9 vex 3059 . . . . . . . . . . . 12  |-  p  e. 
_V
10 inimasn 5271 . . . . . . . . . . . 12  |-  ( p  e.  _V  ->  (
( w  i^i  u
) " { p } )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
119, 10ax-mp 5 . . . . . . . . . . 11  |-  ( ( w  i^i  u )
" { p }
)  =  ( ( w " { p } )  i^i  (
u " { p } ) )
128, 11syl6eqr 2513 . . . . . . . . . 10  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) )
137, 12sylancom 678 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) )
14 imaeq1 5181 . . . . . . . . . . 11  |-  ( x  =  ( w  i^i  u )  ->  (
x " { p } )  =  ( ( w  i^i  u
) " { p } ) )
1514eqeq2d 2471 . . . . . . . . . 10  |-  ( x  =  ( w  i^i  u )  ->  (
( a  i^i  b
)  =  ( x
" { p }
)  <->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) ) )
1615rspcev 3161 . . . . . . . . 9  |-  ( ( ( w  i^i  u
)  e.  U  /\  ( a  i^i  b
)  =  ( ( w  i^i  u )
" { p }
) )  ->  E. x  e.  U  ( a  i^i  b )  =  ( x " { p } ) )
176, 13, 16syl2anc 671 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  E. x  e.  U  ( a  i^i  b )  =  ( x " { p } ) )
18 vex 3059 . . . . . . . . . . 11  |-  a  e. 
_V
1918inex1 4557 . . . . . . . . . 10  |-  ( a  i^i  b )  e. 
_V
20 utopustuq.1 . . . . . . . . . . 11  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 21302 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  (
a  i^i  b )  e.  _V )  ->  (
( a  i^i  b
)  e.  ( N `
 p )  <->  E. x  e.  U  ( a  i^i  b )  =  ( x " { p } ) ) )
2219, 21mpan2 682 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
( a  i^i  b
)  e.  ( N `
 p )  <->  E. x  e.  U  ( a  i^i  b )  =  ( x " { p } ) ) )
2322biimpar 492 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  E. x  e.  U  (
a  i^i  b )  =  ( x " { p } ) )  ->  ( a  i^i  b )  e.  ( N `  p ) )
241, 17, 23syl2anc 671 . . . . . . 7  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  /\  u  e.  U )  /\  b  =  ( u " { p } ) )  ->  ( a  i^i  b )  e.  ( N `  p ) )
25 simp-4l 781 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( U  e.  (UnifOn `  X )  /\  p  e.  X
) )
26 simpllr 774 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  b  e.  ( N `  p ) )
27 vex 3059 . . . . . . . . . 10  |-  b  e. 
_V
2820ustuqtoplem 21302 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
2927, 28mpan2 682 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
3029biimpa 491 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  ( N `  p
) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3125, 26, 30syl2anc 671 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3224, 31r19.29a 2943 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  /\  b  e.  ( N `  p
) )  /\  w  e.  U )  /\  a  =  ( w " { p } ) )  ->  ( a  i^i  b )  e.  ( N `  p ) )
3320ustuqtoplem 21302 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3418, 33mpan2 682 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3534biimpa 491 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
3635adantr 471 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
3732, 36r19.29a 2943 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  (
a  i^i  b )  e.  ( N `  p
) )
3837ralrimiva 2813 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  A. b  e.  ( N `  p
) ( a  i^i  b )  e.  ( N `  p ) )
3938ralrimiva 2813 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  A. a  e.  ( N `  p
) A. b  e.  ( N `  p
) ( a  i^i  b )  e.  ( N `  p ) )
40 fvex 5897 . . . 4  |-  ( N `
 p )  e. 
_V
41 inficl 7964 . . . 4  |-  ( ( N `  p )  e.  _V  ->  ( A. a  e.  ( N `  p ) A. b  e.  ( N `  p )
( a  i^i  b
)  e.  ( N `
 p )  <->  ( fi `  ( N `  p
) )  =  ( N `  p ) ) )
4240, 41ax-mp 5 . . 3  |-  ( A. a  e.  ( N `  p ) A. b  e.  ( N `  p
) ( a  i^i  b )  e.  ( N `  p )  <-> 
( fi `  ( N `  p )
)  =  ( N `
 p ) )
4339, 42sylib 201 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  =  ( N `  p
) )
44 eqimss 3495 . 2  |-  ( ( fi `  ( N `
 p ) )  =  ( N `  p )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
4543, 44syl 17 1  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749   _Vcvv 3056    i^i cin 3414    C_ wss 3415   {csn 3979    |-> cmpt 4474   ran crn 4853   "cima 4855   ` cfv 5600   ficfi 7949  UnifOncust 21262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-fin 7598  df-fi 7950  df-ust 21263
This theorem is referenced by:  ustuqtop  21309  utopsnneiplem  21310
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