Theorem hmeoopn 12961

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

Step	Hyp	Ref	Expression
1		hmeoima 12960	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A e. J ) -> ( F " A ) e. K )
2	1	ex 114	. . 3 |- ( F e. ( J Homeo K ) -> ( A e. J -> ( F " A ) e. K ) )
3	2	adantr 274	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J -> ( F " A ) e. K ) )
4		hmeocn 12955	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
5		cnima 12870	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( F " A ) e. K ) -> ( `' F " ( F " A ) ) e. J )
6	5	ex 114	. . . . 5 |- ( F e. ( J Cn K ) -> ( ( F " A ) e. K -> ( `' F " ( F " A ) ) e. J ) )
7	4, 6	syl 14	. . . 4 |- ( F e. ( J Homeo K ) -> ( ( F " A ) e. K -> ( `' F " ( F " A ) ) e. J ) )
8	7	adantr 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 2165	. . . . . . 7 |- U. K = U. K
11	9, 10	hmeof1o 12959	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
12		f1of1 5431	. . . . . 6 |- ( F : X -1-1-onto-> U. K -> F : X -1-1-> U. K )
13	11, 12	syl 14	. . . . 5 |- ( F e. ( J Homeo K ) -> F : X -1-1-> U. K )
14		f1imacnv 5449	. . . . 5 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
15	13, 14	sylan 281	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
16	15	eleq1d 2235	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( `' F " ( F " A ) ) e. J <-> A e. J ) )
17	8, 16	sylibd 148	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. K -> A e. J ) )
18	3, 17	impbid 128	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   C_ wss 3116   U.cuni 3789   `'ccnv 4603   "cima 4607   -1-1->wf1 5185   -1-1-onto->wf1o 5187  (class class class)co 5842   Cn ccn 12835   Homeochmeo 12950

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-top 12646  df-topon 12659  df-cn 12838  df-hmeo 12951

This theorem is referenced by: (None)