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Theorem ucnprima 20517
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 20516 . . 3  |-  ( ph  ->  E. r  e.  U  ( G " r ) 
C_  W )
75mpt2fun 6386 . . . . 5  |-  Fun  G
8 ustssxp 20439 . . . . . . 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 4711 . . . . . . 7  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
115, 10dmmpt2 6851 . . . . . 6  |-  dom  G  =  ( X  X.  X )
129, 11syl6sseqr 3551 . . . . 5  |-  ( (
ph  /\  r  e.  U )  ->  r  C_ 
dom  G )
13 funimass3 5995 . . . . 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 2970 . . 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 5355 . . . . . 6  |-  ( `' G " W ) 
C_  dom  G
2019, 11sseqtri 3536 . . . . 5  |-  ( `' G " W ) 
C_  ( X  X.  X )
2120a1i 11 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  ( `' G " W ) 
C_  ( X  X.  X ) )
22 ustssel 20440 . . . 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 1228 . . 3  |-  ( (
ph  /\  r  e.  U )  ->  (
r  C_  ( `' G " W )  -> 
( `' G " W )  e.  U
) )
2423rexlimdva 2955 . 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 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   <.cop 4033    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5580   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284  UnifOncust 20434   Cnucucn 20510
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-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-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-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-ust 20435  df-ucn 20511
This theorem is referenced by:  fmucnd  20527
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