Theorem ovolficcss 23238

Description: Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)

Assertion

Ref	Expression
ovolficcss	|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. F ) C_ RR )

Proof of Theorem ovolficcss

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

Step	Hyp	Ref	Expression
1		rnco2 5642	. . 3 |- ran ( [,] o. F ) = ( [,] " ran F )
2		inss2 3834	. . . . . . . . . . . 12 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
3		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( F ` y ) e. ( <_ i^i ( RR X. RR ) ) )
4	2, 3	sseldi 3601	. . . . . . . . . . 11 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( F ` y ) e. ( RR X. RR ) )
5		1st2nd2 7205	. . . . . . . . . . 11 |- ( ( F ` y ) e. ( RR X. RR ) -> ( F ` y ) = <. ( 1st ` ( F ` y ) ) , ( 2nd ` ( F ` y ) ) >. )
6	4, 5	syl 17	. . . . . . . . . 10 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( F ` y ) = <. ( 1st ` ( F ` y ) ) , ( 2nd ` ( F ` y ) ) >. )
7	6	fveq2d 6195	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( [,] ` ( F ` y ) ) = ( [,] ` <. ( 1st ` ( F ` y ) ) , ( 2nd ` ( F ` y ) ) >. ) )
8		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` ( F ` y ) ) [,] ( 2nd ` ( F ` y ) ) ) = ( [,] ` <. ( 1st ` ( F ` y ) ) , ( 2nd ` ( F ` y ) ) >. )
9	7, 8	syl6eqr 2674	. . . . . . . 8 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( [,] ` ( F ` y ) ) = ( ( 1st ` ( F ` y ) ) [,] ( 2nd ` ( F ` y ) ) ) )
10		xp1st 7198	. . . . . . . . . 10 |- ( ( F ` y ) e. ( RR X. RR ) -> ( 1st ` ( F ` y ) ) e. RR )
11	4, 10	syl 17	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( 1st ` ( F ` y ) ) e. RR )
12		xp2nd 7199	. . . . . . . . . 10 |- ( ( F ` y ) e. ( RR X. RR ) -> ( 2nd ` ( F ` y ) ) e. RR )
13	4, 12	syl 17	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( 2nd ` ( F ` y ) ) e. RR )
14		iccssre 12255	. . . . . . . . 9 |- ( ( ( 1st ` ( F ` y ) ) e. RR /\ ( 2nd ` ( F ` y ) ) e. RR ) -> ( ( 1st ` ( F ` y ) ) [,] ( 2nd ` ( F ` y ) ) ) C_ RR )
15	11, 13, 14	syl2anc 693	. . . . . . . 8 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( ( 1st ` ( F ` y ) ) [,] ( 2nd ` ( F ` y ) ) ) C_ RR )
16	9, 15	eqsstrd 3639	. . . . . . 7 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( [,] ` ( F ` y ) ) C_ RR )
17		reex 10027	. . . . . . . 8 |- RR e. _V
18	17	elpw2 4828	. . . . . . 7 |- ( ( [,] ` ( F ` y ) ) e. ~P RR <-> ( [,] ` ( F ` y ) ) C_ RR )
19	16, 18	sylibr 224	. . . . . 6 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ y e. NN ) -> ( [,] ` ( F ` y ) ) e. ~P RR )
20	19	ralrimiva 2966	. . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> A. y e. NN ( [,] ` ( F ` y ) ) e. ~P RR )
21		ffn 6045	. . . . . 6 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> F Fn NN )
22		fveq2 6191	. . . . . . . 8 |- ( x = ( F ` y ) -> ( [,] ` x ) = ( [,] ` ( F ` y ) ) )
23	22	eleq1d 2686	. . . . . . 7 |- ( x = ( F ` y ) -> ( ( [,] ` x ) e. ~P RR <-> ( [,] ` ( F ` y ) ) e. ~P RR ) )
24	23	ralrn 6362	. . . . . 6 |- ( F Fn NN -> ( A. x e. ran F ( [,] ` x ) e. ~P RR <-> A. y e. NN ( [,] ` ( F ` y ) ) e. ~P RR ) )
25	21, 24	syl 17	. . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( A. x e. ran F ( [,] ` x ) e. ~P RR <-> A. y e. NN ( [,] ` ( F ` y ) ) e. ~P RR ) )
26	20, 25	mpbird 247	. . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> A. x e. ran F ( [,] ` x ) e. ~P RR )
27		iccf 12272	. . . . . 6 |- [,] : ( RR* X. RR* ) --> ~P RR*
28		ffun 6048	. . . . . 6 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> Fun [,] )
29	27, 28	ax-mp 5	. . . . 5 |- Fun [,]
30		frn 6053	. . . . . 6 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ran F C_ ( <_ i^i ( RR X. RR ) ) )
31		rexpssxrxp 10084	. . . . . . . 8 |- ( RR X. RR ) C_ ( RR* X. RR* )
32	2, 31	sstri 3612	. . . . . . 7 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR* X. RR* )
33	27	fdmi 6052	. . . . . . 7 |- dom [,] = ( RR* X. RR* )
34	32, 33	sseqtr4i 3638	. . . . . 6 |- ( <_ i^i ( RR X. RR ) ) C_ dom [,]
35	30, 34	syl6ss 3615	. . . . 5 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ran F C_ dom [,] )
36		funimass4 6247	. . . . 5 |- ( ( Fun [,] /\ ran F C_ dom [,] ) -> ( ( [,] " ran F ) C_ ~P RR <-> A. x e. ran F ( [,] ` x ) e. ~P RR ) )
37	29, 35, 36	sylancr 695	. . . 4 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( [,] " ran F ) C_ ~P RR <-> A. x e. ran F ( [,] ` x ) e. ~P RR ) )
38	26, 37	mpbird 247	. . 3 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( [,] " ran F ) C_ ~P RR )
39	1, 38	syl5eqss 3649	. 2 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ran ( [,] o. F ) C_ ~P RR )
40		sspwuni 4611	. 2 |- ( ran ( [,] o. F ) C_ ~P RR <-> U. ran ( [,] o. F ) C_ RR )
41	39, 40	sylib 208	1 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. F ) C_ RR )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   RR*cxr 10073   <_ cle 10075   NNcn 11020   [,]cicc 12178

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  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-nel 2898  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  ovollb2lem  23256  ovollb2  23257  uniiccdif  23346  uniiccvol  23348  uniioombllem3  23353  uniioombllem4  23354  uniioombllem5  23355  uniiccmbl  23358