Theorem xkoopn 20653

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 6343	. . . . . . 7 |- ( R Cn S ) e. _V
2	1	pwex 4600	. . . . . 6 |- ~P ( R Cn S ) e. _V
3		xkoopn.x	. . . . . . . 8 |- X = U. R
4		eqid 2462	. . . . . . . 8 |- { x e. ~P X | ( R ↾t x ) e. Comp } = { x e. ~P X | ( R ↾t x ) e. Comp }
5		eqid 2462	. . . . . . . 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 20649	. . . . . . 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 5758	. . . . . . 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 4562	. . . . 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 7959	. . . . 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 5898	. . . . 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 20030	. . . . 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 3453	. . 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 19985	. . . . . . . 8 |- ( R e. Top -> X e. R )
19		elpw2g 4580	. . . . . . . 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 240	. . . . . 6 |- ( ph -> A e. ~P X )
22		xkoopn.c	. . . . . 6 |- ( ph -> ( R ↾t A ) e. Comp )
23		oveq2 6323	. . . . . . . 8 |- ( x = A -> ( R ↾t x ) = ( R ↾t A ) )
24	23	eleq1d 2524	. . . . . . 7 |- ( x = A -> ( ( R ↾t x ) e. Comp <-> ( R ↾t A ) e. Comp ) )
25	24	elrab 3208	. . . . . 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 675	. . . . 5 |- ( ph -> A e. { x e. ~P X | ( R ↾t x ) e. Comp } )
27		xkoopn.u	. . . . 5 |- ( ph -> U e. S )
28		eqidd 2463	. . . . 5 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
29		imaeq2 5183	. . . . . . . . 9 |- ( k = A -> ( f " k ) = ( f " A ) )
30	29	sseq1d 3471	. . . . . . . 8 |- ( k = A -> ( ( f " k ) C_ v <-> ( f " A ) C_ v ) )
31	30	rabbidv 3048	. . . . . . 7 |- ( k = A -> { f e. ( R Cn S ) | ( f " k ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ v } )
32	31	eqeq2d 2472	. . . . . 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 3466	. . . . . . . 8 |- ( v = U -> ( ( f " A ) C_ v <-> ( f " A ) C_ U ) )
34	33	rabbidv 3048	. . . . . . 7 |- ( v = U -> { f e. ( R Cn S ) | ( f " A ) C_ v } = { f e. ( R Cn S ) | ( f " A ) C_ U } )
35	34	eqeq2d 2472	. . . . . 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 3173	. . . . 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 1276	. . . 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 4568	. . . . 5 |- { f e. ( R Cn S ) | ( f " A ) C_ U } e. _V
39		eqeq1 2466	. . . . . 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 2920	. . . . 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 6433	. . . . 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 3200	. . . 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 217	. . 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 3442	. 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 20651	. . 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 671	. 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 2543	1 |- ( ph -> { f e. ( R Cn S ) | ( f " A ) C_ U } e. ( S ^ko R ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1455   e. wcel 1898   E.wrex 2750   {crab 2753   _Vcvv 3057   C_ wss 3416   ~Pcpw 3963   U.cuni 4212   X. cxp 4851   ran crn 4854   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6315   |-> cmpt2 6317   ficfi 7950   ↾_t crest 15368   topGenctg 15385   Topctop 19966   Cn ccn 20289   Compccmp 20450   ^ko cxko 20625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-1o 7208  df-en 7596  df-fin 7599  df-fi 7951  df-topgen 15391  df-top 19970  df-xko 20627

This theorem is referenced by:  xkouni  20663  xkohaus  20717  xkoptsub  20718  xkoco1cn  20721  xkoco2cn  20722  xkococnlem  20723