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Theorem cnconst2 19543
Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )

Proof of Theorem cnconst2
Dummy variables  x  u  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 5765 . . 3  |-  ( B  e.  Y  ->  ( X  X.  { B }
) : X --> Y )
213ad2ant3 1014 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } ) : X --> Y )
32adantr 465 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
) : X --> Y )
4 simpll3 1032 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  B  e.  Y )
5 simplr 754 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  x  e.  X )
6 fvconst2g 6105 . . . . . . . 8  |-  ( ( B  e.  Y  /\  x  e.  X )  ->  ( ( X  X.  { B } ) `  x )  =  B )
74, 5, 6syl2anc 661 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  (
( X  X.  { B } ) `  x
)  =  B )
87eleq1d 2529 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  (
( ( X  X.  { B } ) `  x )  e.  y  <-> 
B  e.  y ) )
9 simpll1 1030 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  J  e.  (TopOn `  X ) )
10 toponmax 19189 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
119, 10syl 16 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  X  e.  J )
12 simplr 754 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  x  e.  X )
13 df-ima 5005 . . . . . . . . 9  |-  ( ( X  X.  { B } ) " X
)  =  ran  (
( X  X.  { B } )  |`  X )
14 ssid 3516 . . . . . . . . . . . . 13  |-  X  C_  X
15 xpssres 5299 . . . . . . . . . . . . 13  |-  ( X 
C_  X  ->  (
( X  X.  { B } )  |`  X )  =  ( X  X.  { B } ) )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( ( X  X.  { B } )  |`  X )  =  ( X  X.  { B } )
1716rneqi 5220 . . . . . . . . . . 11  |-  ran  (
( X  X.  { B } )  |`  X )  =  ran  ( X  X.  { B }
)
18 rnxpss 5430 . . . . . . . . . . 11  |-  ran  ( X  X.  { B }
)  C_  { B }
1917, 18eqsstri 3527 . . . . . . . . . 10  |-  ran  (
( X  X.  { B } )  |`  X ) 
C_  { B }
20 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  B  e.  y )
2120snssd 4165 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  { B }  C_  y )
2219, 21syl5ss 3508 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  ran  ( ( X  X.  { B } )  |`  X ) 
C_  y )
2313, 22syl5eqss 3541 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  ( ( X  X.  { B }
) " X ) 
C_  y )
24 eleq2 2533 . . . . . . . . . 10  |-  ( u  =  X  ->  (
x  e.  u  <->  x  e.  X ) )
25 imaeq2 5324 . . . . . . . . . . 11  |-  ( u  =  X  ->  (
( X  X.  { B } ) " u
)  =  ( ( X  X.  { B } ) " X
) )
2625sseq1d 3524 . . . . . . . . . 10  |-  ( u  =  X  ->  (
( ( X  X.  { B } ) "
u )  C_  y  <->  ( ( X  X.  { B } ) " X
)  C_  y )
)
2724, 26anbi12d 710 . . . . . . . . 9  |-  ( u  =  X  ->  (
( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
)  <->  ( x  e.  X  /\  ( ( X  X.  { B } ) " X
)  C_  y )
) )
2827rspcev 3207 . . . . . . . 8  |-  ( ( X  e.  J  /\  ( x  e.  X  /\  ( ( X  X.  { B } ) " X )  C_  y
) )  ->  E. u  e.  J  ( x  e.  u  /\  (
( X  X.  { B } ) " u
)  C_  y )
)
2911, 12, 23, 28syl12anc 1221 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  (
y  e.  K  /\  B  e.  y )
)  ->  E. u  e.  J  ( x  e.  u  /\  (
( X  X.  { B } ) " u
)  C_  y )
)
3029expr 615 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  ( B  e.  y  ->  E. u  e.  J  ( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
) ) )
318, 30sylbid 215 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  (
( ( X  X.  { B } ) `  x )  e.  y  ->  E. u  e.  J  ( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
) ) )
3231ralrimiva 2871 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  A. y  e.  K  ( (
( X  X.  { B } ) `  x
)  e.  y  ->  E. u  e.  J  ( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
) ) )
33 simpl1 994 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
34 simpl2 995 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  K  e.  (TopOn `  Y )
)
35 simpr 461 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  x  e.  X )
36 iscnp 19497 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  x  e.  X
)  ->  ( ( X  X.  { B }
)  e.  ( ( J  CnP  K ) `
 x )  <->  ( ( X  X.  { B }
) : X --> Y  /\  A. y  e.  K  ( ( ( X  X.  { B } ) `  x )  e.  y  ->  E. u  e.  J  ( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
) ) ) ) )
3733, 34, 35, 36syl3anc 1223 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  (
( X  X.  { B } )  e.  ( ( J  CnP  K
) `  x )  <->  ( ( X  X.  { B } ) : X --> Y  /\  A. y  e.  K  ( ( ( X  X.  { B } ) `  x
)  e.  y  ->  E. u  e.  J  ( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
) ) ) ) )
383, 32, 37mpbir2and 915 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
)  e.  ( ( J  CnP  K ) `
 x ) )
3938ralrimiva 2871 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  A. x  e.  X  ( X  X.  { B } )  e.  ( ( J  CnP  K ) `  x ) )
40 cncnp 19540 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( X  X.  { B }
)  e.  ( J  Cn  K )  <->  ( ( X  X.  { B }
) : X --> Y  /\  A. x  e.  X  ( X  X.  { B } )  e.  ( ( J  CnP  K
) `  x )
) ) )
41403adant3 1011 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( ( X  X.  { B }
)  e.  ( J  Cn  K )  <->  ( ( X  X.  { B }
) : X --> Y  /\  A. x  e.  X  ( X  X.  { B } )  e.  ( ( J  CnP  K
) `  x )
) ) )
422, 39, 41mpbir2and 915 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   {csn 4020    X. cxp 4990   ran crn 4993    |` cres 4994   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275  TopOnctopon 19155    Cn ccn 19484    CnP ccnp 19485
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 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-topgen 14688  df-top 19159  df-topon 19162  df-cn 19487  df-cnp 19488
This theorem is referenced by:  cnconst  19544  xkoccn  19848  txkgen  19881  cnmptc  19891  pcoptcl  21249  blocni  25382  pl1cn  27559  conpcon  28306  cvmliftphtlem  28388  cvmlift3lem9  28398  cnfdmsn  31175  stoweidlem47  31302
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