MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuqtop2 Structured version   Unicode version

Theorem ustuqtop2 19948
Description: Lemma for ustuqtop 19952 (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 19913 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  u  e.  U )  ->  (
w  i^i  u )  e.  U )
62, 3, 4, 5syl3anc 1219 . . . . . . . . 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 3654 . . . . . . . . . . 11  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
9 vex 3079 . . . . . . . . . . . 12  |-  p  e. 
_V
10 inimasn 5361 . . . . . . . . . . . 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 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 5271 . . . . . . . . . . 11  |-  ( x  =  ( w  i^i  u )  ->  (
x " { p } )  =  ( ( w  i^i  u
) " { p } ) )
1514eqeq2d 2468 . . . . . . . . . 10  |-  ( x  =  ( w  i^i  u )  ->  (
( a  i^i  b
)  =  ( x
" { p }
)  <->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) ) )
1615rspcev 3177 . . . . . . . . 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 3079 . . . . . . . . . . 11  |-  a  e. 
_V
1918inex1 4540 . . . . . . . . . 10  |-  ( a  i^i  b )  e. 
_V
20 utopustuq.1 . . . . . . . . . . 11  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 19945 . . . . . . . . . 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 3079 . . . . . . . . . 10  |-  b  e. 
_V
2820ustuqtoplem 19945 . . . . . . . . . 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 2966 . . . . . 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 19945 . . . . . . . . 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 2966 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  (
a  i^i  b )  e.  ( N `  p
) )
3837ralrimiva 2829 . . . 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 2829 . . 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 5808 . . . 4  |-  ( N `
 p )  e. 
_V
41 inficl 7785 . . . 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 3515 . 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 1370    e. wcel 1758   A.wral 2798   E.wrex 2799   _Vcvv 3076    i^i cin 3434    C_ wss 3435   {csn 3984    |-> cmpt 4457   ran crn 4948   "cima 4950   ` cfv 5525   ficfi 7770  UnifOncust 19905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-fin 7423  df-fi 7771  df-ust 19906
This theorem is referenced by:  ustuqtop  19952  utopsnneiplem  19953
  Copyright terms: Public domain W3C validator