Theorem xkoopn 19825

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 6307	. . . . . . 7 |- ( R Cn S ) e. _V
2	1	pwex 4630	. . . . . 6 |- ~P ( R Cn S ) e. _V
3		xkoopn.x	. . . . . . . 8 |- X = U. R
4		eqid 2467	. . . . . . . 8 |- { x e. ~P X | ( R ↾t x ) e. Comp } = { x e. ~P X | ( R ↾t x ) e. Comp }
5		eqid 2467	. . . . . . . 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 19821	. . . . . . 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 5735	. . . . . . 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 4592	. . . . 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 7875	. . . . 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 5874	. . . . 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 19234	. . . . 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 3513	. . 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 19182	. . . . . . . 8 |- ( R e. Top -> X e. R )
19		elpw2g 4610	. . . . . . . 8 |- ( X e. R -> ( A e. ~P X <-> A C_ X ) )
20	17, 18, 19	3syl 20	. . . . . . 7 |- ( ph -> ( A e. ~P X <-> A C_ X ) )
21	16, 20	mpbird 232	. . . . . 6 |- ( ph -> A e. ~P X )
22		xkoopn.c	. . . . . 6 |- ( ph -> ( R ↾t A ) e. Comp )
23		oveq2 6290	. . . . . . . 8 |- ( x = A -> ( R ↾t x ) = ( R ↾t A ) )
24	23	eleq1d 2536	. . . . . . 7 |- ( x = A -> ( ( R ↾t x ) e. Comp <-> ( R ↾t A ) e. Comp ) )
25	24	elrab 3261	. . . . . 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 664	. . . . 5 |- ( ph -> A e. { x e. ~P X | ( R ↾t x ) e. Comp } )
27		xkoopn.u	. . . . 5 |- ( ph -> U e. S )
28		eqidd 2468	. . . . 5 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
29		imaeq2 5331	. . . . . . . . 9 |- ( k = A -> ( f " k ) = ( f " A ) )
30	29	sseq1d 3531	. . . . . . . 8 |- ( k = A -> ( ( f " k ) C_ v <-> ( f " A ) C_ v ) )
31	30	rabbidv 3105	. . . . . . 7 |- ( k = A -> { f e. ( R Cn S ) | ( f " k ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ v } )
32	31	eqeq2d 2481	. . . . . 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 3526	. . . . . . . 8 |- ( v = U -> ( ( f " A ) C_ v <-> ( f " A ) C_ U ) )
34	33	rabbidv 3105	. . . . . . 7 |- ( v = U -> { f e. ( R Cn S ) | ( f " A ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
35	34	eqeq2d 2481	. . . . . 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 3225	. . . . 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 1228	. . . 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 4598	. . . . 5 |- { f e. ( R Cn S ) | ( f " A ) C_ U } e. _V
39		eqeq1 2471	. . . . . 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 2980	. . . . 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 6394	. . . . 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 3253	. . . 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 212	. . 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 3502	. 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 19823	. . 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 661	. 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 2558	1 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ( S ^ko R ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   X. cxp 4997   ran crn 5000   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   ficfi 7866   ↾_t crest 14672   topGenctg 14689   Topctop 19161   Cn ccn 19491   Compccmp 19652   ^ko cxko 19797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-en 7514  df-fin 7517  df-fi 7867  df-topgen 14695  df-top 19166  df-xko 19799

This theorem is referenced by:  xkouni  19835  xkohaus  19889  xkoptsub  19890  xkoco1cn  19893  xkoco2cn  19894  xkococnlem  19895