Theorem ishmeo 15328

Description: The predicate F is a homeomorphism between topology J and topology K. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
ishmeo	|- ( F e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )

Proof of Theorem ishmeo

Dummy variables f j k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-hmeo 15325	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
2	1	elmpocl 6274	. 2 |- ( F e. ( J Homeo K ) -> ( J e. Top /\ K e. Top ) )
3		df-cn 15212	. . . 4 |- Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } )
4	3	elmpocl 6274	. . 3 |- ( F e. ( J Cn K ) -> ( J e. Top /\ K e. Top ) )
5	4	adantr 276	. 2 |- ( ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) -> ( J e. Top /\ K e. Top ) )
6		hmeofvalg 15327	. . . 4 |- ( ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
7	6	eleq2d 2308	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( F e. ( J Homeo K ) <-> F e. { f e. ( J Cn K ) | `' f e. ( K Cn J ) } ) )
8		cnveq 4949	. . . . 5 |- ( f = F -> `' f = `' F )
9	8	eleq1d 2307	. . . 4 |- ( f = F -> ( `' f e. ( K Cn J ) <-> `' F e. ( K Cn J ) ) )
10	9	elrab 2982	. . 3 |- ( F e. { f e. ( J Cn K ) | `' f e. ( K Cn J ) } <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )
11	7, 10	bitrdi 196	. 2 |- ( ( J e. Top /\ K e. Top ) -> ( F e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) ) )
12	2, 5, 11	pm5.21nii 716	1 |- ( F e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   U.cuni 3930   `'ccnv 4768   "cima 4772  (class class class)co 6075   ^m cmap 6912   Topctop 15021   Cn ccn 15209   Homeochmeo 15324

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-top 15022  df-topon 15035  df-cn 15212  df-hmeo 15325

This theorem is referenced by:  hmeocn  15329  hmeocnvcn  15330  hmeocnv  15331  hmeores  15339  hmeoco  15340  idhmeo  15341  txhmeo  15343  txswaphmeo  15345  cnrehmeocntop  15634