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Theorem ucnprima 21228
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 21227 . . 3  |-  ( ph  ->  E. r  e.  U  ( G " r ) 
C_  W )
75mpt2fun 6412 . . . . 5  |-  Fun  G
8 ustssxp 21150 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  r  e.  U )  ->  r  C_  ( X  X.  X
) )
91, 8sylan 473 . . . . . 6  |-  ( (
ph  /\  r  e.  U )  ->  r  C_  ( X  X.  X
) )
10 opex 4686 . . . . . . 7  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
115, 10dmmpt2 6877 . . . . . 6  |-  dom  G  =  ( X  X.  X )
129, 11syl6sseqr 3517 . . . . 5  |-  ( (
ph  /\  r  e.  U )  ->  r  C_ 
dom  G )
13 funimass3 6013 . . . . 5  |-  ( ( Fun  G  /\  r  C_ 
dom  G )  -> 
( ( G "
r )  C_  W  <->  r 
C_  ( `' G " W ) ) )
147, 12, 13sylancr 667 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  (
( G " r
)  C_  W  <->  r  C_  ( `' G " W ) ) )
1514rexbidva 2943 . . 3  |-  ( ph  ->  ( E. r  e.  U  ( G "
r )  C_  W  <->  E. r  e.  U  r 
C_  ( `' G " W ) ) )
166, 15mpbid 213 . 2  |-  ( ph  ->  E. r  e.  U  r  C_  ( `' G " W ) )
171adantr 466 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  U  e.  (UnifOn `  X )
)
18 simpr 462 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  r  e.  U )
19 cnvimass 5208 . . . . . 6  |-  ( `' G " W ) 
C_  dom  G
2019, 11sseqtri 3502 . . . . 5  |-  ( `' G " W ) 
C_  ( X  X.  X )
2120a1i 11 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  ( `' G " W ) 
C_  ( X  X.  X ) )
22 ustssel 21151 . . . 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 1264 . . 3  |-  ( (
ph  /\  r  e.  U )  ->  (
r  C_  ( `' G " W )  -> 
( `' G " W )  e.  U
) )
2423rexlimdva 2924 . 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783    C_ wss 3442   <.cop 4008    X. cxp 4852   `'ccnv 4853   dom cdm 4854   "cima 4857   Fun wfun 5595   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307  UnifOncust 21145   Cnucucn 21221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-ust 21146  df-ucn 21222
This theorem is referenced by:  fmucnd  21238
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