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Theorem ucnprima 20910
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 20909 . . 3  |-  ( ph  ->  E. r  e.  U  ( G " r ) 
C_  W )
75mpt2fun 6403 . . . . 5  |-  Fun  G
8 ustssxp 20832 . . . . . . 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 4720 . . . . . . 7  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
115, 10dmmpt2 6869 . . . . . 6  |-  dom  G  =  ( X  X.  X )
129, 11syl6sseqr 3546 . . . . 5  |-  ( (
ph  /\  r  e.  U )  ->  r  C_ 
dom  G )
13 funimass3 6004 . . . . 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 2965 . . 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 5367 . . . . . 6  |-  ( `' G " W ) 
C_  dom  G
2019, 11sseqtri 3531 . . . . 5  |-  ( `' G " W ) 
C_  ( X  X.  X )
2120a1i 11 . . . 4  |-  ( (
ph  /\  r  e.  U )  ->  ( `' G " W ) 
C_  ( X  X.  X ) )
22 ustssel 20833 . . . 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 2949 . 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 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   <.cop 4038    X. cxp 5006   `'ccnv 5007   dom cdm 5008   "cima 5011   Fun wfun 5588   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  UnifOncust 20827   Cnucucn 20903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-ust 20828  df-ucn 20904
This theorem is referenced by:  fmucnd  20920
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