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Theorem cnconst2 19657
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 5764 . . 3  |-  ( B  e.  Y  ->  ( X  X.  { B }
) : X --> Y )
213ad2ant3 1020 . 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 1038 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  B  e.  Y )
5 simplr 755 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  x  e.  X )
6 fvconst2g 6109 . . . . . . . 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 2512 . . . . . 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 1036 . . . . . . . . 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 19302 . . . . . . . . 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 755 . . . . . . . 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 5002 . . . . . . . . 9  |-  ( ( X  X.  { B } ) " X
)  =  ran  (
( X  X.  { B } )  |`  X )
14 ssid 3508 . . . . . . . . . . . . 13  |-  X  C_  X
15 xpssres 5298 . . . . . . . . . . . . 13  |-  ( X 
C_  X  ->  (
( X  X.  { B } )  |`  X )  =  ( X  X.  { B } ) )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( ( X  X.  { B } )  |`  X )  =  ( X  X.  { B } )
1716rneqi 5219 . . . . . . . . . . 11  |-  ran  (
( X  X.  { B } )  |`  X )  =  ran  ( X  X.  { B }
)
18 rnxpss 5429 . . . . . . . . . . 11  |-  ran  ( X  X.  { B }
)  C_  { B }
1917, 18eqsstri 3519 . . . . . . . . . 10  |-  ran  (
( X  X.  { B } )  |`  X ) 
C_  { B }
20 simprr 757 . . . . . . . . . . 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 4160 . . . . . . . . . 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 3500 . . . . . . . . 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 3533 . . . . . . . 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 2516 . . . . . . . . . 10  |-  ( u  =  X  ->  (
x  e.  u  <->  x  e.  X ) )
25 imaeq2 5323 . . . . . . . . . . 11  |-  ( u  =  X  ->  (
( X  X.  { B } ) " u
)  =  ( ( X  X.  { B } ) " X
) )
2625sseq1d 3516 . . . . . . . . . 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 3196 . . . . . . . 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 1227 . . . . . . 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 2857 . . . 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 1000 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
34 simpl2 1001 . . . . 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 19611 . . . . 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 1229 . . . 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 922 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
)  e.  ( ( J  CnP  K ) `
 x ) )
3938ralrimiva 2857 . 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 19654 . . 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 1017 . 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 922 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   {csn 4014    X. cxp 4987   ran crn 4990    |` cres 4991   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281  TopOnctopon 19268    Cn ccn 19598    CnP ccnp 19599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-topgen 14718  df-top 19272  df-topon 19275  df-cn 19601  df-cnp 19602
This theorem is referenced by:  cnconst  19658  xkoccn  19993  txkgen  20026  cnmptc  20036  pcoptcl  21394  blocni  25592  pl1cn  27810  conpcon  28553  cvmliftphtlem  28635  cvmlift3lem9  28645  cnfdmsn  31591  stoweidlem47  31718
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