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Theorem mapdiscn 25630
Description: Any mapping whose domain is associated to the discrete topology is continuous. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 7-Apr-2015.)
Hypothesis
Ref Expression
mapdiscn2.2  |-  B  = 
U. J
Assertion
Ref Expression
mapdiscn  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F  e.  ( ~P A  Cn  J ) )

Proof of Theorem mapdiscn
StepHypRef Expression
1 simp2 956 . . 3  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F : A --> B )
2 mapdiscn2.2 . . . . . 6  |-  B  = 
U. J
32topopn 16668 . . . . 5  |-  ( J  e.  Top  ->  B  e.  J )
433ad2ant1 976 . . . 4  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  B  e.  J )
5 simp3 957 . . . 4  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  A  e.  D )
6 elmapg 6801 . . . 4  |-  ( ( B  e.  J  /\  A  e.  D )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
74, 5, 6syl2anc 642 . . 3  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
81, 7mpbird 223 . 2  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F  e.  ( B  ^m  A ) )
9 simp1 955 . . . 4  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  J  e.  Top )
102toptopon 16687 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  B ) )
119, 10sylib 188 . . 3  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  J  e.  (TopOn `  B ) )
12 cndis 17035 . . 3  |-  ( ( A  e.  D  /\  J  e.  (TopOn `  B
) )  ->  ( ~P A  Cn  J
)  =  ( B  ^m  A ) )
135, 11, 12syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  ( ~P A  Cn  J )  =  ( B  ^m  A ) )
148, 13eleqtrrd 2373 1  |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F  e.  ( ~P A  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   ~Pcpw 3638   U.cuni 3843   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  usptoplem  25649  istopx  25650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973
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