Theorem hmeofvalg 15114

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 15007	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
2		rabexg 4238	. . 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 6037	. . . 4 |- ( ( j = J /\ k = K ) -> ( j Cn k ) = ( J Cn K ) )
5		oveq12 6037	. . . . . 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 2301	. . . 4 |- ( ( j = J /\ k = K ) -> ( `' f e. ( k Cn j ) <-> `' f e. ( K Cn J ) ) )
8	4, 7	rabeqbidv 2798	. . 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 15112	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
10	8, 9	ovmpoga 6161	. 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 1375	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 1398   e. wcel 2202   {crab 2515   _Vcvv 2803   `'ccnv 4730  (class class class)co 6028   Topctop 14808   Cn ccn 14996   Homeochmeo 15111

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cn 14999  df-hmeo 15112

This theorem is referenced by:  ishmeo  15115