MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dnsconst Structured version   Unicode version

Theorem dnsconst 19109
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 5925). (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 754 . . 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 18977 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
5 ffn 5662 . . 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 996 . . 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 753 . . . . . 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 994 . . . . . 6  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  P  e.  Y )
103t1sncld 19057 . . . . . 6  |-  ( ( K  e.  Fre  /\  P  e.  Y )  ->  { P }  e.  ( Clsd `  K )
)
118, 9, 10syl2anc 661 . . . . 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 18999 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  { P }  e.  (
Clsd `  K )
)  ->  ( `' F " { P }
)  e.  ( Clsd `  J ) )
131, 11, 12syl2anc 661 . . . 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 995 . . . 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 18803 . . . 4  |-  ( ( ( `' F " { P } )  e.  ( Clsd `  J
)  /\  A  C_  ( `' F " { P } ) )  -> 
( ( cls `  J
) `  A )  C_  ( `' F " { P } ) )
1613, 14, 15syl2anc 661 . . 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 3494 . 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 6045 . 2  |-  ( F : X --> { P } 
<->  ( F  Fn  X  /\  X  C_  ( `' F " { P } ) ) )
196, 17, 18sylanbrc 664 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3431   {csn 3980   U.cuni 4194   `'ccnv 4942   "cima 4946    Fn wfn 5516   -->wf 5517   ` cfv 5521  (class class class)co 6195   Clsdccld 18747   clsccl 18749    Cn ccn 18955   Frect1 19038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-top 18630  df-topon 18633  df-cld 18750  df-cls 18752  df-cn 18958  df-t1 19045
This theorem is referenced by:  ipasslem8  24384
  Copyright terms: Public domain W3C validator