Theorem hmeoopn 14754

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 14753	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A e. J ) -> ( F " A ) e. K )
2	1	ex 115	. . 3 |- ( F e. ( J Homeo K ) -> ( A e. J -> ( F " A ) e. K ) )
3	2	adantr 276	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J -> ( F " A ) e. K ) )
4		hmeocn 14748	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
5		cnima 14663	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( F " A ) e. K ) -> ( `' F " ( F " A ) ) e. J )
6	5	ex 115	. . . . 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 276	. . 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 2204	. . . . . . 7 |- U. K = U. K
11	9, 10	hmeof1o 14752	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
12		f1of1 5520	. . . . . 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 5538	. . . . 5 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
15	13, 14	sylan 283	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
16	15	eleq1d 2273	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( `' F " ( F " A ) ) e. J <-> A e. J ) )
17	8, 16	sylibd 149	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. K -> A e. J ) )
18	3, 17	impbid 129	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   C_ wss 3165   U.cuni 3849   `'ccnv 4673   "cima 4677   -1-1->wf1 5267   -1-1-onto->wf1o 5269  (class class class)co 5943   Cn ccn 14628   Homeochmeo 14743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-top 14441  df-topon 14454  df-cn 14631  df-hmeo 14744

This theorem is referenced by: (None)