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Theorem ustuqtop2 18225
Description: Lemma for ustuqtop 18229 (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 747 . . . . . . . 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 749 . . . . . . . . . 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 744 . . . . . . . . . 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 732 . . . . . . . . . 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 18190 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U  /\  u  e.  U )  ->  (
w  i^i  u )  e.  U )
62, 3, 4, 5syl3anc 1184 . . . . . . . . 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 736 . . . . . . . . . 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 3497 . . . . . . . . . . 11  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
9 vex 2919 . . . . . . . . . . . 12  |-  p  e. 
_V
10 inimasn 5248 . . . . . . . . . . . 12  |-  ( p  e.  _V  ->  (
( w  i^i  u
) " { p } )  =  ( ( w " {
p } )  i^i  ( u " {
p } ) ) )
119, 10ax-mp 8 . . . . . . . . . . 11  |-  ( ( w  i^i  u )
" { p }
)  =  ( ( w " { p } )  i^i  (
u " { p } ) )
128, 11syl6eqr 2454 . . . . . . . . . 10  |-  ( ( a  =  ( w
" { p }
)  /\  b  =  ( u " {
p } ) )  ->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) )
137, 12sylancom 649 . . . . . . . . 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 5157 . . . . . . . . . . 11  |-  ( x  =  ( w  i^i  u )  ->  (
x " { p } )  =  ( ( w  i^i  u
) " { p } ) )
1514eqeq2d 2415 . . . . . . . . . 10  |-  ( x  =  ( w  i^i  u )  ->  (
( a  i^i  b
)  =  ( x
" { p }
)  <->  ( a  i^i  b )  =  ( ( w  i^i  u
) " { p } ) ) )
1615rspcev 3012 . . . . . . . . 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 643 . . . . . . . 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 2919 . . . . . . . . . . 11  |-  a  e. 
_V
1918inex1 4304 . . . . . . . . . 10  |-  ( a  i^i  b )  e. 
_V
20 utopustuq.1 . . . . . . . . . . 11  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
2120ustuqtoplem 18222 . . . . . . . . . 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 653 . . . . . . . . 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 472 . . . . . . . 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 643 . . . . . . 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 743 . . . . . . . 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 736 . . . . . . . 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 2919 . . . . . . . . . 10  |-  b  e. 
_V
2820ustuqtoplem 18222 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  _V )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
2927, 28mpan2 653 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
b  e.  ( N `
 p )  <->  E. u  e.  U  b  =  ( u " {
p } ) ) )
3029biimpa 471 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  b  e.  ( N `  p
) )  ->  E. u  e.  U  b  =  ( u " {
p } ) )
3125, 26, 30syl2anc 643 . . . . . . 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 2810 . . . . . 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 18222 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  _V )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3418, 33mpan2 653 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  (
a  e.  ( N `
 p )  <->  E. w  e.  U  a  =  ( w " {
p } ) ) )
3534biimpa 471 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. w  e.  U  a  =  ( w " {
p } ) )
3635adantr 452 . . . . . 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 2810 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  e.  ( N `  p ) )  /\  b  e.  ( N `  p
) )  ->  (
a  i^i  b )  e.  ( N `  p
) )
3837ralrimiva 2749 . . . 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 2749 . . 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 5701 . . . 4  |-  ( N `
 p )  e. 
_V
41 inficl 7388 . . . 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 8 . . 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 189 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  =  ( N `  p
) )
44 eqimss 3360 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   {csn 3774    e. cmpt 4226   ran crn 4838   "cima 4840   ` cfv 5413   ficfi 7373  UnifOncust 18182
This theorem is referenced by:  ustuqtop  18229  utopsnneiplem  18230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-ust 18183
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