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Theorem dnsconst 20046
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 5981). (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 753 . . 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 19914 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
5 ffn 5713 . . 3  |-  ( F : X --> Y  ->  F  Fn  X )
61, 4, 53syl 20 . 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 1002 . . 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 751 . . . . . 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 1000 . . . . . 6  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  P  e.  Y )
103t1sncld 19994 . . . . . 6  |-  ( ( K  e.  Fre  /\  P  e.  Y )  ->  { P }  e.  ( Clsd `  K )
)
118, 9, 10syl2anc 659 . . . . 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 19936 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  { P }  e.  (
Clsd `  K )
)  ->  ( `' F " { P }
)  e.  ( Clsd `  J ) )
131, 11, 12syl2anc 659 . . . 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 1001 . . . 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 19740 . . . 4  |-  ( ( ( `' F " { P } )  e.  ( Clsd `  J
)  /\  A  C_  ( `' F " { P } ) )  -> 
( ( cls `  J
) `  A )  C_  ( `' F " { P } ) )
1613, 14, 15syl2anc 659 . . 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 3524 . 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 6110 . 2  |-  ( F : X --> { P } 
<->  ( F  Fn  X  /\  X  C_  ( `' F " { P } ) ) )
196, 17, 18sylanbrc 662 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   {csn 4016   U.cuni 4235   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Clsdccld 19684   clsccl 19686    Cn ccn 19892   Frect1 19975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19566  df-topon 19569  df-cld 19687  df-cls 19689  df-cn 19895  df-t1 19982
This theorem is referenced by:  ipasslem8  25950
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