Theorem hmeofvalg 14471

Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
hmeofvalg	|- ( ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )

Distinct variable groups:   f, J   f, K

Proof of Theorem hmeofvalg

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

Step	Hyp	Ref	Expression
1		cnovex 14364	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
2		rabexg 4172	. . 3 |- ( ( J Cn K ) e. _V -> { f e. ( J Cn K ) | `' f e. ( K Cn J ) } e. _V )
3	1, 2	syl 14	. 2 |- ( ( J e. Top /\ K e. Top ) -> { f e. ( J Cn K ) | `' f e. ( K Cn J ) } e. _V )
4		oveq12 5927	. . . 4 |- ( ( j = J /\ k = K ) -> ( j Cn k ) = ( J Cn K ) )
5		oveq12 5927	. . . . . 6 |- ( ( k = K /\ j = J ) -> ( k Cn j ) = ( K Cn J ) )
6	5	ancoms 268	. . . . 5 |- ( ( j = J /\ k = K ) -> ( k Cn j ) = ( K Cn J ) )
7	6	eleq2d 2263	. . . 4 |- ( ( j = J /\ k = K ) -> ( `' f e. ( k Cn j ) <-> `' f e. ( K Cn J ) ) )
8	4, 7	rabeqbidv 2755	. . 3 |- ( ( j = J /\ k = K ) -> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
9		df-hmeo 14469	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
10	8, 9	ovmpoga 6048	. 2 |- ( ( J e. Top /\ K e. Top /\ { f e. ( J Cn K ) | `' f e. ( K Cn J ) } e. _V ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
11	3, 10	mpd3an3 1349	1 |- ( ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760   `'ccnv 4658  (class class class)co 5918   Topctop 14165   Cn ccn 14353   Homeochmeo 14468

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-top 14166  df-topon 14179  df-cn 14356  df-hmeo 14469

This theorem is referenced by:  ishmeo  14472