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Theorem hmeoopn 13066
Description: Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1  |-  X  = 
U. J
Assertion
Ref Expression
hmeoopn  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K
) )

Proof of Theorem hmeoopn
StepHypRef Expression
1 hmeoima 13065 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
21ex 114 . . 3  |-  ( F  e.  ( J Homeo K )  ->  ( A  e.  J  ->  ( F
" A )  e.  K ) )
32adantr 274 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  ->  ( F " A )  e.  K ) )
4 hmeocn 13060 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
5 cnima 12975 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  K )  -> 
( `' F "
( F " A
) )  e.  J
)
65ex 114 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
74, 6syl 14 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  ( ( F " A )  e.  K  ->  ( `' F " ( F " A ) )  e.  J ) )
87adantr 274 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
9 hmeoopn.1 . . . . . . 7  |-  X  = 
U. J
10 eqid 2170 . . . . . . 7  |-  U. K  =  U. K
119, 10hmeof1o 13064 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
12 f1of1 5439 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1311, 12syl 14 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F : X -1-1-> U. K )
14 f1imacnv 5457 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1513, 14sylan 281 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1615eleq1d 2239 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  J  <->  A  e.  J ) )
178, 16sylibd 148 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  K  ->  A  e.  J )
)
183, 17impbid 128 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141    C_ wss 3121   U.cuni 3794   `'ccnv 4608   "cima 4612   -1-1->wf1 5193   -1-1-onto->wf1o 5195  (class class class)co 5851    Cn ccn 12940   Homeochmeo 13055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6625  df-top 12751  df-topon 12764  df-cn 12943  df-hmeo 13056
This theorem is referenced by: (None)
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