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Theorem dnsconst 17442
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 5848). (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 732 . . 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 17310 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
5 ffn 5591 . . 3  |-  ( F : X --> Y  ->  F  Fn  X )
61, 4, 53syl 19 . 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 965 . . 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 731 . . . . . 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 963 . . . . . 6  |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  (
( cls `  J
) `  A )  =  X ) )  ->  P  e.  Y )
103t1sncld 17390 . . . . . 6  |-  ( ( K  e.  Fre  /\  P  e.  Y )  ->  { P }  e.  ( Clsd `  K )
)
118, 9, 10syl2anc 643 . . . . 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 17332 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  { P }  e.  (
Clsd `  K )
)  ->  ( `' F " { P }
)  e.  ( Clsd `  J ) )
131, 11, 12syl2anc 643 . . . 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 964 . . . 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 17136 . . . 4  |-  ( ( ( `' F " { P } )  e.  ( Clsd `  J
)  /\  A  C_  ( `' F " { P } ) )  -> 
( ( cls `  J
) `  A )  C_  ( `' F " { P } ) )
1613, 14, 15syl2anc 643 . . 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 3383 . 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 5955 . 2  |-  ( F : X --> { P } 
<->  ( F  Fn  X  /\  X  C_  ( `' F " { P } ) ) )
196, 17, 18sylanbrc 646 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   {csn 3814   U.cuni 4015   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Clsdccld 17080   clsccl 17082    Cn ccn 17288   Frect1 17371
This theorem is referenced by:  ipasslem8  22338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cld 17083  df-cls 17085  df-cn 17291  df-t1 17378
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