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Theorem ucnprima 19855
Description: The preimage by a uniformly continuous function  F of an entourage  W of  Y is an entourage of  X. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
ucnprima.2  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
ucnprima.3  |-  ( ph  ->  F  e.  ( U Cnu V ) )
ucnprima.4  |-  ( ph  ->  W  e.  V )
ucnprima.5  |-  G  =  ( x  e.  X ,  y  e.  X  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)
Assertion
Ref Expression
ucnprima  |-  ( ph  ->  ( `' G " W )  e.  U
)
Distinct variable groups:    x, y, F    x, X, y    x, G, y    x, U, y   
x, V    x, W, y    x, Y    ph, x, y
Allowed substitution hints:    V( y)    Y( y)

Proof of Theorem ucnprima
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 ucnprima.1 . . . 4  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
2 ucnprima.2 . . . 4  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
3 ucnprima.3 . . . 4  |-  ( ph  ->  F  e.  ( U Cnu V ) )
4 ucnprima.4 . . . 4  |-  ( ph  ->  W  e.  V )
5 ucnprima.5 . . . 4  |-  G  =  ( x  e.  X ,  y  e.  X  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)
61, 2, 3, 4, 5ucnima 19854 . . 3  |-  ( ph  ->  E. r  e.  U  ( G " r ) 
C_  W )
75mpt2fun 6190 . . . . 5  |-  Fun  G
8 ustssxp 19777 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  r  e.  U )  ->  r  C_  ( X  X.  X
) )
91, 8sylan 471 . . . . . 6  |-  ( (
ph  /\  r  e.  U )  ->  r  C_  ( X  X.  X
) )
10 opex 4554 . . . . . . 7  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
115, 10dmmpt2 6642 . . . . . 6  |-  dom  G  =  ( X  X.  X )
129, 11syl6sseqr 3401 . . . . 5  |-  ( (
ph  /\  r  e.  U )  ->  r  C_ 
dom  G )
13 funimass3 5817 . . . . 5  |-  ( ( Fun  G  /\  r  C_ 
dom  G )  -> 
( ( G "
r )  C_  W  <->  r 
C_  ( `' G " W ) ) )
147, 12, 13sylancr 663 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  (
( G " r
)  C_  W  <->  r  C_  ( `' G " W ) ) )
1514rexbidva 2730 . . 3  |-  ( ph  ->  ( E. r  e.  U  ( G "
r )  C_  W  <->  E. r  e.  U  r 
C_  ( `' G " W ) ) )
166, 15mpbid 210 . 2  |-  ( ph  ->  E. r  e.  U  r  C_  ( `' G " W ) )
171adantr 465 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  U  e.  (UnifOn `  X )
)
18 simpr 461 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  r  e.  U )
19 cnvimass 5187 . . . . . 6  |-  ( `' G " W ) 
C_  dom  G
2019, 11sseqtri 3386 . . . . 5  |-  ( `' G " W ) 
C_  ( X  X.  X )
2120a1i 11 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  ( `' G " W ) 
C_  ( X  X.  X ) )
22 ustssel 19778 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  r  e.  U  /\  ( `' G " W ) 
C_  ( X  X.  X ) )  -> 
( r  C_  ( `' G " W )  ->  ( `' G " W )  e.  U
) )
2317, 18, 21, 22syl3anc 1218 . . 3  |-  ( (
ph  /\  r  e.  U )  ->  (
r  C_  ( `' G " W )  -> 
( `' G " W )  e.  U
) )
2423rexlimdva 2839 . 2  |-  ( ph  ->  ( E. r  e.  U  r  C_  ( `' G " W )  ->  ( `' G " W )  e.  U
) )
2516, 24mpd 15 1  |-  ( ph  ->  ( `' G " W )  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714    C_ wss 3326   <.cop 3881    X. cxp 4836   `'ccnv 4837   dom cdm 4838   "cima 4841   Fun wfun 5410   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091  UnifOncust 19772   Cnucucn 19848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-map 7214  df-ust 19773  df-ucn 19849
This theorem is referenced by:  fmucnd  19865
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