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Theorem hauseqcn 26261
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 18803 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
42, 3syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
5 dmin 5043 . . . . . 6  |-  dom  ( F  i^i  G )  C_  ( dom  F  i^i  dom  G )
6 eqid 2441 . . . . . . . . . 10  |-  U. J  =  U. J
7 eqid 2441 . . . . . . . . . 10  |-  U. K  =  U. K
86, 7cnf 18809 . . . . . . . . 9  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
9 fdm 5560 . . . . . . . . 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 18809 . . . . . . . . 9  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
13 fdm 5560 . . . . . . . . 9  |-  ( G : U. J --> U. K  ->  dom  G  =  U. J )
1411, 12, 133syl 20 . . . . . . . 8  |-  ( ph  ->  dom  G  =  U. J )
1510, 14ineq12d 3550 . . . . . . 7  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( U. J  i^i  U. J ) )
16 inidm 3556 . . . . . . 7  |-  ( U. J  i^i  U. J )  =  U. J
1715, 16syl6eq 2489 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  U. J )
185, 17syl5sseq 3401 . . . . 5  |-  ( ph  ->  dom  ( F  i^i  G )  C_  U. J )
19 hauseqcn.e . . . . . 6  |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )
20 ffn 5556 . . . . . . . 8  |-  ( F : U. J --> U. K  ->  F  Fn  U. J
)
212, 8, 203syl 20 . . . . . . 7  |-  ( ph  ->  F  Fn  U. J
)
22 ffn 5556 . . . . . . . 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 3399 . . . . . . 7  |-  ( ph  ->  A  C_  U. J )
26 fnreseql 5810 . . . . . . 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 1213 . . . . . 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 18617 . . . . 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 1213 . . . 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 19178 . . . . 5  |-  ( ph  ->  dom  ( F  i^i  G )  e.  ( Clsd `  J ) )
34 cldcls 18605 . . . . 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 3395 . . 3  |-  ( ph  ->  X  C_  dom  ( F  i^i  G ) )
371, 36syl5eqssr 3398 . 2  |-  ( ph  ->  U. J  C_  dom  ( F  i^i  G ) )
38 fneqeql2 5809 . . 3  |-  ( ( F  Fn  U. J  /\  G  Fn  U. J
)  ->  ( F  =  G  <->  U. J  C_  dom  ( F  i^i  G ) ) )
3921, 23, 38syl2anc 656 . 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 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   U.cuni 4088   dom cdm 4836    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Topctop 18457   Clsdccld 18579   clsccl 18581    Cn ccn 18787   Hauscha 18871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465  df-cld 18582  df-cls 18584  df-cn 18790  df-haus 18878  df-tx 19094
This theorem is referenced by:  rrhre  26383
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