Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hauseqcn Structured version   Unicode version

Theorem hauseqcn 27702
Description: In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
Hypotheses
Ref Expression
hauseqcn.x  |-  X  = 
U. J
hauseqcn.k  |-  ( ph  ->  K  e.  Haus )
hauseqcn.f  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
hauseqcn.g  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
hauseqcn.e  |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )
hauseqcn.a  |-  ( ph  ->  A  C_  X )
hauseqcn.c  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
Assertion
Ref Expression
hauseqcn  |-  ( ph  ->  F  =  G )

Proof of Theorem hauseqcn
StepHypRef Expression
1 hauseqcn.x . . 3  |-  X  = 
U. J
2 hauseqcn.f . . . . . 6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 cntop1 19609 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
42, 3syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
5 dmin 5216 . . . . . 6  |-  dom  ( F  i^i  G )  C_  ( dom  F  i^i  dom  G )
6 eqid 2467 . . . . . . . . . 10  |-  U. J  =  U. J
7 eqid 2467 . . . . . . . . . 10  |-  U. K  =  U. K
86, 7cnf 19615 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
9 fdm 5741 . . . . . . . . 9  |-  ( F : U. J --> U. K  ->  dom  F  =  U. J )
102, 8, 93syl 20 . . . . . . . 8  |-  ( ph  ->  dom  F  =  U. J )
11 hauseqcn.g . . . . . . . . 9  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
126, 7cnf 19615 . . . . . . . . 9  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
13 fdm 5741 . . . . . . . . 9  |-  ( G : U. J --> U. K  ->  dom  G  =  U. J )
1411, 12, 133syl 20 . . . . . . . 8  |-  ( ph  ->  dom  G  =  U. J )
1510, 14ineq12d 3706 . . . . . . 7  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( U. J  i^i  U. J ) )
16 inidm 3712 . . . . . . 7  |-  ( U. J  i^i  U. J )  =  U. J
1715, 16syl6eq 2524 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  U. J )
185, 17syl5sseq 3557 . . . . 5  |-  ( ph  ->  dom  ( F  i^i  G )  C_  U. J )
19 hauseqcn.e . . . . . 6  |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )
20 ffn 5737 . . . . . . . 8  |-  ( F : U. J --> U. K  ->  F  Fn  U. J
)
212, 8, 203syl 20 . . . . . . 7  |-  ( ph  ->  F  Fn  U. J
)
22 ffn 5737 . . . . . . . 8  |-  ( G : U. J --> U. K  ->  G  Fn  U. J
)
2311, 12, 223syl 20 . . . . . . 7  |-  ( ph  ->  G  Fn  U. J
)
24 hauseqcn.a . . . . . . . 8  |-  ( ph  ->  A  C_  X )
2524, 1syl6sseq 3555 . . . . . . 7  |-  ( ph  ->  A  C_  U. J )
26 fnreseql 5998 . . . . . . 7  |-  ( ( F  Fn  U. J  /\  G  Fn  U. J  /\  A  C_  U. J
)  ->  ( ( F  |`  A )  =  ( G  |`  A )  <-> 
A  C_  dom  ( F  i^i  G ) ) )
2721, 23, 25, 26syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( F  |`  A )  =  ( G  |`  A )  <->  A 
C_  dom  ( F  i^i  G ) ) )
2819, 27mpbid 210 . . . . 5  |-  ( ph  ->  A  C_  dom  ( F  i^i  G ) )
296clsss 19423 . . . . 5  |-  ( ( J  e.  Top  /\  dom  ( F  i^i  G
)  C_  U. J  /\  A  C_  dom  ( F  i^i  G ) )  ->  ( ( cls `  J ) `  A
)  C_  ( ( cls `  J ) `  dom  ( F  i^i  G
) ) )
304, 18, 28, 29syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( cls `  J
) `  A )  C_  ( ( cls `  J
) `  dom  ( F  i^i  G ) ) )
31 hauseqcn.c . . . 4  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
32 hauseqcn.k . . . . . 6  |-  ( ph  ->  K  e.  Haus )
3332, 2, 11hauseqlcld 20015 . . . . 5  |-  ( ph  ->  dom  ( F  i^i  G )  e.  ( Clsd `  J ) )
34 cldcls 19411 . . . . 5  |-  ( dom  ( F  i^i  G
)  e.  ( Clsd `  J )  ->  (
( cls `  J
) `  dom  ( F  i^i  G ) )  =  dom  ( F  i^i  G ) )
3533, 34syl 16 . . . 4  |-  ( ph  ->  ( ( cls `  J
) `  dom  ( F  i^i  G ) )  =  dom  ( F  i^i  G ) )
3630, 31, 353sstr3d 3551 . . 3  |-  ( ph  ->  X  C_  dom  ( F  i^i  G ) )
371, 36syl5eqssr 3554 . 2  |-  ( ph  ->  U. J  C_  dom  ( F  i^i  G ) )
38 fneqeql2 5997 . . 3  |-  ( ( F  Fn  U. J  /\  G  Fn  U. J
)  ->  ( F  =  G  <->  U. J  C_  dom  ( F  i^i  G ) ) )
3921, 23, 38syl2anc 661 . 2  |-  ( ph  ->  ( F  =  G  <->  U. J  C_  dom  ( F  i^i  G ) ) )
4037, 39mpbird 232 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   U.cuni 4251   dom cdm 5005    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Topctop 19263   Clsdccld 19385   clsccl 19387    Cn ccn 19593   Hauscha 19677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-topgen 14716  df-top 19268  df-bases 19270  df-topon 19271  df-cld 19388  df-cls 19390  df-cn 19596  df-haus 19684  df-tx 19931
This theorem is referenced by:  rrhre  27824
  Copyright terms: Public domain W3C validator