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Theorem ustuqtop2 20480
Description: Lemma for ustuqtop 20484 (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 769 . . . . . . . 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 771 . . . . . . . . . 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 766 . . . . . . . . . 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 754 . . . . . . . . . 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 20445 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  u  e.  U )  ->  (
w  i^i  u )  e.  U )
62, 3, 4, 5syl3anc 1228 . . . . . . . . 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 758 . . . . . . . . . 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 3695 . . . . . . . . . . 11  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
9 vex 3116 . . . . . . . . . . . 12  |-  p  e. 
_V
10 inimasn 5421 . . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) )
137, 12sylancom 667 . . . . . . . . 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 5330 . . . . . . . . . . 11  |-  ( x  =  ( w  i^i  u )  ->  (
x " { p } )  =  ( ( w  i^i  u
) " { p } ) )
1514eqeq2d 2481 . . . . . . . . . 10  |-  ( x  =  ( w  i^i  u )  ->  (
( a  i^i  b
)  =  ( x
" { p }
)  <->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) ) )
1615rspcev 3214 . . . . . . . . 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 661 . . . . . . . 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 3116 . . . . . . . . . . 11  |-  a  e. 
_V
1918inex1 4588 . . . . . . . . . 10  |-  ( a  i^i  b )  e. 
_V
20 utopustuq.1 . . . . . . . . . . 11  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 20477 . . . . . . . . . 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 671 . . . . . . . . 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 485 . . . . . . . 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 661 . . . . . . 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 765 . . . . . . . 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 758 . . . . . . . 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 3116 . . . . . . . . . 10  |-  b  e. 
_V
2820ustuqtoplem 20477 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
2927, 28mpan2 671 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
3029biimpa 484 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  ( N `  p
) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3125, 26, 30syl2anc 661 . . . . . . 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 3003 . . . . . 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 20477 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3418, 33mpan2 671 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3534biimpa 484 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
3635adantr 465 . . . . . 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 3003 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  (
a  i^i  b )  e.  ( N `  p
) )
3837ralrimiva 2878 . . . 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 2878 . . 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 5874 . . . 4  |-  ( N `
 p )  e. 
_V
41 inficl 7881 . . . 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 196 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  =  ( N `  p
) )
44 eqimss 3556 . 2  |-  ( ( fi `  ( N `
 p ) )  =  ( N `  p )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
4543, 44syl 16 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   {csn 4027    |-> cmpt 4505   ran crn 5000   "cima 5002   ` cfv 5586   ficfi 7866  UnifOncust 20437
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 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-ust 20438
This theorem is referenced by:  ustuqtop  20484  utopsnneiplem  20485
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