Theorem hmeofvalg 15026

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 14919	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
2		rabexg 4233	. . 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 6026	. . . 4 |- ( ( j = J /\ k = K ) -> ( j Cn k ) = ( J Cn K ) )
5		oveq12 6026	. . . . . 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 2797	. . 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 15024	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
10	8, 9	ovmpoga 6150	. 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 1374	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 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   `'ccnv 4724  (class class class)co 6017   Topctop 14720   Cn ccn 14908   Homeochmeo 15023

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cn 14911  df-hmeo 15024

This theorem is referenced by:  ishmeo  15027