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Theorem dnsconst 17122
Description: If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that  ( ( cls `  J ) `  A )  =  X means " A is dense in  X " and  A  C_  ( `' F " { P } ) means " F is constant on  A " (see funconstss 5659). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
dnsconst.1  |-  X  = 
U. J
dnsconst.2  |-  Y  = 
U. K
Assertion
Ref Expression
dnsconst  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  F : X --> { P } )

Proof of Theorem dnsconst
StepHypRef Expression
1 simplr 731 . . 3  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  F  e.  ( J  Cn  K ) )
2 dnsconst.1 . . . 4  |-  X  = 
U. J
3 dnsconst.2 . . . 4  |-  Y  = 
U. K
42, 3cnf 16992 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
5 ffn 5405 . . 3  |-  ( F : X --> Y  ->  F  Fn  X )
61, 4, 53syl 18 . 2  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  F  Fn  X )
7 simpr3 963 . . 3  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  -> 
( ( cls `  J
) `  A )  =  X )
8 simpll 730 . . . . . 6  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  K  e.  Fre )
9 simpr1 961 . . . . . 6  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  P  e.  Y )
103t1sncld 17070 . . . . . 6  |-  ( ( K  e.  Fre  /\  P  e.  Y )  ->  { P }  e.  ( Clsd `  K )
)
118, 9, 10syl2anc 642 . . . . 5  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  { P }  e.  (
Clsd `  K )
)
12 cnclima 17013 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  { P }  e.  (
Clsd `  K )
)  ->  ( `' F " { P }
)  e.  ( Clsd `  J ) )
131, 11, 12syl2anc 642 . . . 4  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  -> 
( `' F " { P } )  e.  ( Clsd `  J
) )
14 simpr2 962 . . . 4  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  A  C_  ( `' F " { P } ) )
152clsss2 16825 . . . 4  |-  ( ( ( `' F " { P } )  e.  ( Clsd `  J
)  /\  A  C_  ( `' F " { P } ) )  -> 
( ( cls `  J
) `  A )  C_  ( `' F " { P } ) )
1613, 14, 15syl2anc 642 . . 3  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  -> 
( ( cls `  J
) `  A )  C_  ( `' F " { P } ) )
177, 16eqsstr3d 3226 . 2  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  X  C_  ( `' F " { P } ) )
18 fconst3 5751 . 2  |-  ( F : X --> { P } 
<->  ( F  Fn  X  /\  X  C_  ( `' F " { P } ) ) )
196, 17, 18sylanbrc 645 1  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  F : X --> { P } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   U.cuni 3843   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Clsdccld 16769   clsccl 16771    Cn ccn 16970   Frect1 17051
This theorem is referenced by:  ipasslem8  21431
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-rep 4147  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-reu 2563  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-int 3879  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cld 16772  df-cls 16774  df-cn 16973  df-t1 17058
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