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Theorem cstucnd 21079
Description: A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypotheses
Ref Expression
cstucnd.1  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
cstucnd.2  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
cstucnd.3  |-  ( ph  ->  A  e.  Y )
Assertion
Ref Expression
cstucnd  |-  ( ph  ->  ( X  X.  { A } )  e.  ( U Cnu V ) )

Proof of Theorem cstucnd
Dummy variables  s 
r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cstucnd.3 . . 3  |-  ( ph  ->  A  e.  Y )
2 fconst6g 5757 . . 3  |-  ( A  e.  Y  ->  ( X  X.  { A }
) : X --> Y )
31, 2syl 17 . 2  |-  ( ph  ->  ( X  X.  { A } ) : X --> Y )
4 cstucnd.1 . . . . . 6  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
54adantr 463 . . . . 5  |-  ( (
ph  /\  s  e.  V )  ->  U  e.  (UnifOn `  X )
)
6 ustne0 21008 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
75, 6syl 17 . . . 4  |-  ( (
ph  /\  s  e.  V )  ->  U  =/=  (/) )
8 cstucnd.2 . . . . . . . . . 10  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
98ad3antrrr 728 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  V  e.  (UnifOn `  Y
) )
10 simpllr 761 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
s  e.  V )
111ad3antrrr 728 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  A  e.  Y )
12 ustref 21013 . . . . . . . . 9  |-  ( ( V  e.  (UnifOn `  Y )  /\  s  e.  V  /\  A  e.  Y )  ->  A
s A )
139, 10, 11, 12syl3anc 1230 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  A s A )
14 simprl 756 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
15 fvconst2g 6105 . . . . . . . . 9  |-  ( ( A  e.  Y  /\  x  e.  X )  ->  ( ( X  X.  { A } ) `  x )  =  A )
1611, 14, 15syl2anc 659 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( X  X.  { A } ) `  x )  =  A )
17 simprr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
18 fvconst2g 6105 . . . . . . . . 9  |-  ( ( A  e.  Y  /\  y  e.  X )  ->  ( ( X  X.  { A } ) `  y )  =  A )
1911, 17, 18syl2anc 659 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( X  X.  { A } ) `  y )  =  A )
2013, 16, 193brtr4d 4425 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( X  X.  { A } ) `  x ) s ( ( X  X.  { A } ) `  y
) )
2120a1d 25 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x r y  ->  ( ( X  X.  { A }
) `  x )
s ( ( X  X.  { A }
) `  y )
) )
2221ralrimivva 2825 . . . . 5  |-  ( ( ( ph  /\  s  e.  V )  /\  r  e.  U )  ->  A. x  e.  X  A. y  e.  X  ( x
r y  ->  (
( X  X.  { A } ) `  x
) s ( ( X  X.  { A } ) `  y
) ) )
2322reximdva0 3750 . . . 4  |-  ( ( ( ph  /\  s  e.  V )  /\  U  =/=  (/) )  ->  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x
r y  ->  (
( X  X.  { A } ) `  x
) s ( ( X  X.  { A } ) `  y
) ) )
247, 23mpdan 666 . . 3  |-  ( (
ph  /\  s  e.  V )  ->  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x
r y  ->  (
( X  X.  { A } ) `  x
) s ( ( X  X.  { A } ) `  y
) ) )
2524ralrimiva 2818 . 2  |-  ( ph  ->  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( ( X  X.  { A }
) `  x )
s ( ( X  X.  { A }
) `  y )
) )
26 isucn 21073 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y )
)  ->  ( ( X  X.  { A }
)  e.  ( U Cnu V )  <->  ( ( X  X.  { A }
) : X --> Y  /\  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  (
x r y  -> 
( ( X  X.  { A } ) `  x ) s ( ( X  X.  { A } ) `  y
) ) ) ) )
274, 8, 26syl2anc 659 . 2  |-  ( ph  ->  ( ( X  X.  { A } )  e.  ( U Cnu V )  <->  ( ( X  X.  { A }
) : X --> Y  /\  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  (
x r y  -> 
( ( X  X.  { A } ) `  x ) s ( ( X  X.  { A } ) `  y
) ) ) ) )
283, 25, 27mpbir2and 923 1  |-  ( ph  ->  ( X  X.  { A } )  e.  ( U Cnu V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   (/)c0 3738   {csn 3972   class class class wbr 4395    X. cxp 4821   -->wf 5565   ` cfv 5569  (class class class)co 6278  UnifOncust 20994   Cnucucn 21070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-ust 20995  df-ucn 21071
This theorem is referenced by: (None)
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