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Theorem dnsconst 17106
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 5643). (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 16976 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
5 ffn 5389 . . 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 17054 . . . . . 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 16997 . . . . 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 16809 . . . 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 3213 . 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 5735 . 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 1623    e. wcel 1684    C_ wss 3152   {csn 3640   U.cuni 3827   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Clsdccld 16753   clsccl 16755    Cn ccn 16954   Frect1 17035
This theorem is referenced by:  ipasslem8  21415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957  df-t1 17042
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