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Theorem ustuqtop2 20914
Description: Lemma for ustuqtop 20918 (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 753 . . . . . . . . . 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 20879 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  u  e.  U )  ->  (
w  i^i  u )  e.  U )
62, 3, 4, 5syl3anc 1226 . . . . . . . . 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 3681 . . . . . . . . . . 11  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
9 vex 3109 . . . . . . . . . . . 12  |-  p  e. 
_V
10 inimasn 5408 . . . . . . . . . . . 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 665 . . . . . . . . 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 5320 . . . . . . . . . . 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 3207 . . . . . . . . 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 659 . . . . . . . 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 3109 . . . . . . . . . . 11  |-  a  e. 
_V
1918inex1 4578 . . . . . . . . . 10  |-  ( a  i^i  b )  e. 
_V
20 utopustuq.1 . . . . . . . . . . 11  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 20911 . . . . . . . . . 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 669 . . . . . . . . 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 483 . . . . . . . 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 659 . . . . . . 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 3109 . . . . . . . . . 10  |-  b  e. 
_V
2820ustuqtoplem 20911 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
2927, 28mpan2 669 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
3029biimpa 482 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  ( N `  p
) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3125, 26, 30syl2anc 659 . . . . . . 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 2996 . . . . . 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 20911 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3418, 33mpan2 669 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3534biimpa 482 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
3635adantr 463 . . . . . 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 2996 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  (
a  i^i  b )  e.  ( N `  p
) )
3837ralrimiva 2868 . . . 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 2868 . . 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 5858 . . . 4  |-  ( N `
 p )  e. 
_V
41 inficl 7877 . . . 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 3541 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   {csn 4016    |-> cmpt 4497   ran crn 4989   "cima 4991   ` cfv 5570   ficfi 7862  UnifOncust 20871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-fin 7513  df-fi 7863  df-ust 20872
This theorem is referenced by:  ustuqtop  20918  utopsnneiplem  20919
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