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Theorem cnconst2 17027
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 5446 . . 3  |-  ( B  e.  Y  ->  ( X  X.  { B }
) : X --> Y )
213ad2ant3 978 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } ) : X --> Y )
32adantr 451 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
) : X --> Y )
4 simpll3 996 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  B  e.  Y )
5 simplr 731 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  x  e.  X )
6 fvconst2g 5743 . . . . . . . 8  |-  ( ( B  e.  Y  /\  x  e.  X )  ->  ( ( X  X.  { B } ) `  x )  =  B )
74, 5, 6syl2anc 642 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  (
( X  X.  { B } ) `  x
)  =  B )
87eleq1d 2362 . . . . . 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 994 . . . . . . . . 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 16682 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
119, 10syl 15 . . . . . . . 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 731 . . . . . . . 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 4718 . . . . . . . . 9  |-  ( ( X  X.  { B } ) " X
)  =  ran  (
( X  X.  { B } )  |`  X )
14 ssid 3210 . . . . . . . . . . . . 13  |-  X  C_  X
15 xpssres 5005 . . . . . . . . . . . . 13  |-  ( X 
C_  X  ->  (
( X  X.  { B } )  |`  X )  =  ( X  X.  { B } ) )
1614, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( ( X  X.  { B } )  |`  X )  =  ( X  X.  { B } )
1716rneqi 4921 . . . . . . . . . . 11  |-  ran  (
( X  X.  { B } )  |`  X )  =  ran  ( X  X.  { B }
)
18 rnxpss 5124 . . . . . . . . . . 11  |-  ran  ( X  X.  { B }
)  C_  { B }
1917, 18eqsstri 3221 . . . . . . . . . 10  |-  ran  (
( X  X.  { B } )  |`  X ) 
C_  { B }
20 simprr 733 . . . . . . . . . . 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 3776 . . . . . . . . . 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 3203 . . . . . . . . 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 3235 . . . . . . . 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 2357 . . . . . . . . . 10  |-  ( u  =  X  ->  (
x  e.  u  <->  x  e.  X ) )
25 imaeq2 5024 . . . . . . . . . . 11  |-  ( u  =  X  ->  (
( X  X.  { B } ) " u
)  =  ( ( X  X.  { B } ) " X
) )
2625sseq1d 3218 . . . . . . . . . 10  |-  ( u  =  X  ->  (
( ( X  X.  { B } ) "
u )  C_  y  <->  ( ( X  X.  { B } ) " X
)  C_  y )
)
2724, 26anbi12d 691 . . . . . . . . 9  |-  ( u  =  X  ->  (
( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
)  <->  ( x  e.  X  /\  ( ( X  X.  { B } ) " X
)  C_  y )
) )
2827rspcev 2897 . . . . . . . 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 1180 . . . . . . 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 598 . . . . . 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 206 . . . . 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 2639 . . . 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 958 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
34 simpl2 959 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  K  e.  (TopOn `  Y )
)
35 simpr 447 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  x  e.  X )
36 iscnp 16983 . . . . 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 1182 . . . 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 888 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
)  e.  ( ( J  CnP  K ) `
 x ) )
3938ralrimiva 2639 . 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 17025 . . 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 975 . 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 888 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   {csn 3653    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648    Cn ccn 16970    CnP ccnp 16971
This theorem is referenced by:  cnconst  17028  xkoccn  17329  txkgen  17362  cnmptc  17372  pcoptcl  18535  blocni  21399  conpcon  23781  cvmliftphtlem  23863  cvmlift3lem9  23873  stoweidlem47  27899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-topgen 13360  df-top 16652  df-topon 16655  df-cn 16973  df-cnp 16974
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