Theorem hmeofval 21561

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
hmeofval	|- ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) }

Distinct variable groups:   f, J   f, K

Proof of Theorem hmeofval

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

Step	Hyp	Ref	Expression
1		oveq12 6659	. . . 4 |- ( ( j = J /\ k = K ) -> ( j Cn k ) = ( J Cn K ) )
2		oveq12 6659	. . . . . 6 |- ( ( k = K /\ j = J ) -> ( k Cn j ) = ( K Cn J ) )
3	2	ancoms 469	. . . . 5 |- ( ( j = J /\ k = K ) -> ( k Cn j ) = ( K Cn J ) )
4	3	eleq2d 2687	. . . 4 |- ( ( j = J /\ k = K ) -> ( `' f e. ( k Cn j ) <-> `' f e. ( K Cn J ) ) )
5	1, 4	rabeqbidv 3195	. . 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 ) } )
6		df-hmeo 21558	. . 3 |- Homeo = ( j e. Top , k e. Top |-> { f e. ( j Cn k ) | `' f e. ( k Cn j ) } )
7		ovex 6678	. . . 4 |- ( J Cn K ) e. _V
8	7	rabex 4813	. . 3 |- { f e. ( J Cn K ) | `' f e. ( K Cn J ) } e. _V
9	5, 6, 8	ovmpt2a 6791	. 2 |- ( ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
10	6	mpt2ndm0 6875	. . 3 |- ( -. ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = (/) )
11		cntop1 21044	. . . . . . . 8 |- ( f e. ( J Cn K ) -> J e. Top )
12		cntop2 21045	. . . . . . . 8 |- ( f e. ( J Cn K ) -> K e. Top )
13	11, 12	jca 554	. . . . . . 7 |- ( f e. ( J Cn K ) -> ( J e. Top /\ K e. Top ) )
14	13	a1d 25	. . . . . 6 |- ( f e. ( J Cn K ) -> ( `' f e. ( K Cn J ) -> ( J e. Top /\ K e. Top ) ) )
15	14	con3rr3 151	. . . . 5 |- ( -. ( J e. Top /\ K e. Top ) -> ( f e. ( J Cn K ) -> -. `' f e. ( K Cn J ) ) )
16	15	ralrimiv 2965	. . . 4 |- ( -. ( J e. Top /\ K e. Top ) -> A. f e. ( J Cn K ) -. `' f e. ( K Cn J ) )
17		rabeq0 3957	. . . 4 |- ( { f e. ( J Cn K ) | `' f e. ( K Cn J ) } = (/) <-> A. f e. ( J Cn K ) -. `' f e. ( K Cn J ) )
18	16, 17	sylibr 224	. . 3 |- ( -. ( J e. Top /\ K e. Top ) -> { f e. ( J Cn K ) | `' f e. ( K Cn J ) } = (/) )
19	10, 18	eqtr4d 2659	. 2 |- ( -. ( J e. Top /\ K e. Top ) -> ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) } )
20	9, 19	pm2.61i 176	1 |- ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) }

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   (/)c0 3915   `'ccnv 5113  (class class class)co 6650   Topctop 20698   Cn ccn 21028   Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558

This theorem is referenced by:  ishmeo  21562