Theorem xkoopn 21392

Description: A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
xkoopn.x	|- X = U. R
xkoopn.r	|- ( ph -> R e. Top )
xkoopn.s	|- ( ph -> S e. Top )
xkoopn.a	|- ( ph -> A C_ X )
xkoopn.c	|- ( ph -> ( R ↾t A ) e. Comp )
xkoopn.u	|- ( ph -> U e. S )

Assertion

Ref	Expression
xkoopn	|- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ( S ^ko R ) )

Distinct variable groups:   A, f   R, f   S, f   U, f

Allowed substitution hints:   ph( f)   X( f)

Proof of Theorem xkoopn

Dummy variables k v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . . . . 7 |- ( R Cn S ) e. _V
2	1	pwex 4848	. . . . . 6 |- ~P ( R Cn S ) e. _V
3		xkoopn.x	. . . . . . . 8 |- X = U. R
4		eqid 2622	. . . . . . . 8 |- { x e. ~P X | ( R ↾t x ) e. Comp } = { x e. ~P X | ( R ↾t x ) e. Comp }
5		eqid 2622	. . . . . . . 8 |- ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) = ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } )
6	3, 4, 5	xkotf 21388	. . . . . . 7 |- ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) : ( { x e. ~P X | ( R ↾t x ) e. Comp } X. S ) --> ~P ( R Cn S )
7		frn 6053	. . . . . . 7 |- ( ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) : ( { x e. ~P X | ( R ↾t x ) e. Comp } X. S ) --> ~P ( R Cn S ) -> ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) C_ ~P ( R Cn S ) )
8	6, 7	ax-mp 5	. . . . . 6 |- ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) C_ ~P ( R Cn S )
9	2, 8	ssexi 4803	. . . . 5 |- ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) e. _V
10		ssfii 8325	. . . . 5 |- ( ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) e. _V -> ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) C_ ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) )
11	9, 10	ax-mp 5	. . . 4 |- ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) C_ ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
12		fvex 6201	. . . . 5 |- ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) e. _V
13		bastg 20770	. . . . 5 |- ( ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) e. _V -> ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) C_ ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) ) )
14	12, 13	ax-mp 5	. . . 4 |- ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) C_ ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) )
15	11, 14	sstri 3612	. . 3 |- ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) C_ ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) )
16		xkoopn.a	. . . . . . 7 |- ( ph -> A C_ X )
17		xkoopn.r	. . . . . . . 8 |- ( ph -> R e. Top )
18	3	topopn 20711	. . . . . . . 8 |- ( R e. Top -> X e. R )
19		elpw2g 4827	. . . . . . . 8 |- ( X e. R -> ( A e. ~P X <-> A C_ X ) )
20	17, 18, 19	3syl 18	. . . . . . 7 |- ( ph -> ( A e. ~P X <-> A C_ X ) )
21	16, 20	mpbird 247	. . . . . 6 |- ( ph -> A e. ~P X )
22		xkoopn.c	. . . . . 6 |- ( ph -> ( R ↾t A ) e. Comp )
23		oveq2 6658	. . . . . . . 8 |- ( x = A -> ( R ↾t x ) = ( R ↾t A ) )
24	23	eleq1d 2686	. . . . . . 7 |- ( x = A -> ( ( R ↾t x ) e. Comp <-> ( R ↾t A ) e. Comp ) )
25	24	elrab 3363	. . . . . 6 |- ( A e. { x e. ~P X | ( R ↾t x ) e. Comp } <-> ( A e. ~P X /\ ( R ↾t A ) e. Comp ) )
26	21, 22, 25	sylanbrc 698	. . . . 5 |- ( ph -> A e. { x e. ~P X | ( R ↾t x ) e. Comp } )
27		xkoopn.u	. . . . 5 |- ( ph -> U e. S )
28		eqidd 2623	. . . . 5 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
29		imaeq2 5462	. . . . . . . . 9 |- ( k = A -> ( f " k ) = ( f " A ) )
30	29	sseq1d 3632	. . . . . . . 8 |- ( k = A -> ( ( f " k ) C_ v <-> ( f " A ) C_ v ) )
31	30	rabbidv 3189	. . . . . . 7 |- ( k = A -> { f e. ( R Cn S ) | ( f " k ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ v } )
32	31	eqeq2d 2632	. . . . . 6 |- ( k = A -> ( { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ v } ) )
33		sseq2 3627	. . . . . . . 8 |- ( v = U -> ( ( f " A ) C_ v <-> ( f " A ) C_ U ) )
34	33	rabbidv 3189	. . . . . . 7 |- ( v = U -> { f e. ( R Cn S ) | ( f " A ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
35	34	eqeq2d 2632	. . . . . 6 |- ( v = U -> ( { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ v } <-> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ U } ) )
36	32, 35	rspc2ev 3324	. . . . 5 |- ( ( A e. { x e. ~P X | ( R ↾t x ) e. Comp } /\ U e. S /\ { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ U } ) -> E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } )
37	26, 27, 28, 36	syl3anc 1326	. . . 4 |- ( ph -> E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } )
38	1	rabex 4813	. . . . 5 |- { f e. ( R Cn S ) | ( f " A ) C_ U } e. _V
39		eqeq1 2626	. . . . . 6 |- ( y = { f e. ( R Cn S ) | ( f " A ) C_ U } -> ( y = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
40	39	2rexbidv 3057	. . . . 5 |- ( y = { f e. ( R Cn S ) | ( f " A ) C_ U } -> ( E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S y = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
41	5	rnmpt2 6770	. . . . 5 |- ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) = { y | E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S y = { f e. ( R Cn S ) | ( f " k ) C_ v } }
42	38, 40, 41	elab2 3354	. . . 4 |- ( { f e. ( R Cn S ) | ( f " A ) C_ U } e. ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) <-> E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " k ) C_ v } )
43	37, 42	sylibr 224	. . 3 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
44	15, 43	sseldi 3601	. 2 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) ) )
45		xkoopn.s	. . 3 |- ( ph -> S e. Top )
46	3, 4, 5	xkoval 21390	. . 3 |- ( ( R e. Top /\ S e. Top ) -> ( S ^ko R ) = ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) ) )
47	17, 45, 46	syl2anc 693	. 2 |- ( ph -> ( S ^ko R ) = ( topGen ` ( fi ` ran ( k e. { x e. ~P X | ( R ↾t x ) e. Comp } , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) ) ) )
48	44, 47	eleqtrrd 2704	1 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ( S ^ko R ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ficfi 8316   ↾_t crest 16081   topGenctg 16098   Topctop 20698   Cn ccn 21028   Compccmp 21189   ^ko cxko 21364

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-3or 1038  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-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-top 20699  df-xko 21366

This theorem is referenced by:  xkouni  21402  xkohaus  21456  xkoptsub  21457  xkoco1cn  21460  xkoco2cn  21461  xkococnlem  21462