Theorem hmeofvalg 13097

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 12990	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
2		rabexg 4132	. . 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 5862	. . . 4 |- ( ( j = J /\ k = K ) -> ( j Cn k ) = ( J Cn K ) )
5		oveq12 5862	. . . . . 6 |- ( ( k = K /\ j = J ) -> ( k Cn j ) = ( K Cn J ) )
6	5	ancoms 266	. . . . 5 |- ( ( j = J /\ k = K ) -> ( k Cn j ) = ( K Cn J ) )
7	6	eleq2d 2240	. . . 4 |- ( ( j = J /\ k = K ) -> ( `' f e. ( k Cn j ) <-> `' f e. ( K Cn J ) ) )
8	4, 7	rabeqbidv 2725	. . 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 13095	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
10	8, 9	ovmpoga 5982	. 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 1333	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 103   = wceq 1348   e. wcel 2141   {crab 2452   _Vcvv 2730   `'ccnv 4610  (class class class)co 5853   Topctop 12789   Cn ccn 12979   Homeochmeo 13094

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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cn 12982  df-hmeo 13095

This theorem is referenced by:  ishmeo  13098