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Theorem cstucnd 20517
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 5767 . . 3  |-  ( A  e.  Y  ->  ( X  X.  { A }
) : X --> Y )
31, 2syl 16 . 2  |-  ( ph  ->  ( X  X.  { A } ) : X --> Y )
4 cstucnd.1 . . . . . 6  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
54adantr 465 . . . . 5  |-  ( (
ph  /\  s  e.  V )  ->  U  e.  (UnifOn `  X )
)
6 ustne0 20446 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
75, 6syl 16 . . . 4  |-  ( (
ph  /\  s  e.  V )  ->  U  =/=  (/) )
8 cstucnd.2 . . . . . . . . . 10  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
98ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  V  e.  (UnifOn `  Y
) )
10 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
s  e.  V )
111ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  A  e.  Y )
12 ustref 20451 . . . . . . . . 9  |-  ( ( V  e.  (UnifOn `  Y )  /\  s  e.  V  /\  A  e.  Y )  ->  A
s A )
139, 10, 11, 12syl3anc 1223 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  A s A )
14 simprl 755 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
15 fvconst2g 6107 . . . . . . . . 9  |-  ( ( A  e.  Y  /\  x  e.  X )  ->  ( ( X  X.  { A } ) `  x )  =  A )
1611, 14, 15syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( X  X.  { A } ) `  x )  =  A )
17 simprr 756 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
18 fvconst2g 6107 . . . . . . . . 9  |-  ( ( A  e.  Y  /\  y  e.  X )  ->  ( ( X  X.  { A } ) `  y )  =  A )
1911, 17, 18syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  V )  /\  r  e.  U
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( X  X.  { A } ) `  y )  =  A )
2013, 16, 193brtr4d 4472 . . . . . . 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 2880 . . . . 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 3791 . . . 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 668 . . 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 2873 . 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 20511 . . 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 661 . 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 915 1  |-  ( ph  ->  ( X  X.  { A } )  e.  ( U Cnu V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   (/)c0 3780   {csn 4022   class class class wbr 4442    X. cxp 4992   -->wf 5577   ` cfv 5581  (class class class)co 6277  UnifOncust 20432   Cnucucn 20508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-ust 20433  df-ucn 20509
This theorem is referenced by: (None)
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