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Theorem cnconst2 20297
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 5789 . . 3  |-  ( B  e.  Y  ->  ( X  X.  { B }
) : X --> Y )
213ad2ant3 1028 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } ) : X --> Y )
32adantr 466 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
) : X --> Y )
4 simpll3 1046 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  B  e.  Y )
5 simplr 760 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  x  e.  X )
6 fvconst2g 6133 . . . . . . . 8  |-  ( ( B  e.  Y  /\  x  e.  X )  ->  ( ( X  X.  { B } ) `  x )  =  B )
74, 5, 6syl2anc 665 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  /\  x  e.  X )  /\  y  e.  K )  ->  (
( X  X.  { B } ) `  x
)  =  B )
87eleq1d 2491 . . . . . 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 1044 . . . . . . . . 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 19941 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
119, 10syl 17 . . . . . . . 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 760 . . . . . . . 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 4866 . . . . . . . . 9  |-  ( ( X  X.  { B } ) " X
)  =  ran  (
( X  X.  { B } )  |`  X )
14 ssid 3483 . . . . . . . . . . . . 13  |-  X  C_  X
15 xpssres 5158 . . . . . . . . . . . . 13  |-  ( X 
C_  X  ->  (
( X  X.  { B } )  |`  X )  =  ( X  X.  { B } ) )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( ( X  X.  { B } )  |`  X )  =  ( X  X.  { B } )
1716rneqi 5080 . . . . . . . . . . 11  |-  ran  (
( X  X.  { B } )  |`  X )  =  ran  ( X  X.  { B }
)
18 rnxpss 5288 . . . . . . . . . . 11  |-  ran  ( X  X.  { B }
)  C_  { B }
1917, 18eqsstri 3494 . . . . . . . . . 10  |-  ran  (
( X  X.  { B } )  |`  X ) 
C_  { B }
20 simprr 764 . . . . . . . . . . 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 4145 . . . . . . . . . 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 3475 . . . . . . . . 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 3508 . . . . . . . 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 2496 . . . . . . . . . 10  |-  ( u  =  X  ->  (
x  e.  u  <->  x  e.  X ) )
25 imaeq2 5183 . . . . . . . . . . 11  |-  ( u  =  X  ->  (
( X  X.  { B } ) " u
)  =  ( ( X  X.  { B } ) " X
) )
2625sseq1d 3491 . . . . . . . . . 10  |-  ( u  =  X  ->  (
( ( X  X.  { B } ) "
u )  C_  y  <->  ( ( X  X.  { B } ) " X
)  C_  y )
)
2724, 26anbi12d 715 . . . . . . . . 9  |-  ( u  =  X  ->  (
( x  e.  u  /\  ( ( X  X.  { B } ) "
u )  C_  y
)  <->  ( x  e.  X  /\  ( ( X  X.  { B } ) " X
)  C_  y )
) )
2827rspcev 3182 . . . . . . . 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 1262 . . . . . . 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 618 . . . . . 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 218 . . . . 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 2836 . . . 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 1008 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
34 simpl2 1009 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  K  e.  (TopOn `  Y )
)
35 simpr 462 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  x  e.  X )
36 iscnp 20251 . . . . 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 1264 . . . 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 930 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  /\  x  e.  X )  ->  ( X  X.  { B }
)  e.  ( ( J  CnP  K ) `
 x ) )
3938ralrimiva 2836 . 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 20294 . . 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 1025 . 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 930 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772    C_ wss 3436   {csn 3998    X. cxp 4851   ran crn 4854    |` cres 4855   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305  TopOnctopon 19916    Cn ccn 20238    CnP ccnp 20239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7485  df-topgen 15341  df-top 19919  df-topon 19921  df-cn 20241  df-cnp 20242
This theorem is referenced by:  cnconst  20298  xkoccn  20632  txkgen  20665  cnmptc  20675  pcoptcl  22050  blocni  26444  pl1cn  28769  conpcon  29966  cvmliftphtlem  30048  cvmlift3lem9  30058  cnfdmsn  37699  stoweidlem47  37848
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