Theorem ioorf 19496

Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypothesis

Ref	Expression
ioorf.1	|- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) )

Assertion

Ref	Expression
ioorf	|- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Proof of Theorem ioorf

Dummy variables a b w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioorf.1	. 2 |- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) )
2		ioof 11033	. . . 4 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5620	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4		ovelrn 6251	. . . 4 |- ( (,) Fn ( RR* X. RR* ) -> ( x e. ran (,) <-> E. a e. RR* E. b e. RR* x = ( a (,) b ) ) )
5	2, 3, 4	mp2b 10	. . 3 |- ( x e. ran (,) <-> E. a e. RR* E. b e. RR* x = ( a (,) b ) )
6		0le0 10112	. . . . . . . . 9 |- 0 <_ 0
7		df-br 4238	. . . . . . . . 9 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
8	6, 7	mpbi 201	. . . . . . . 8 |- <. 0 , 0 >. e. <_
9		0xr 9162	. . . . . . . . 9 |- 0 e. RR*
10		opelxpi 4939	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> <. 0 , 0 >. e. ( RR* X. RR* ) )
11	9, 9, 10	mp2an 655	. . . . . . . 8 |- <. 0 , 0 >. e. ( RR* X. RR* )
12		elin 3516	. . . . . . . 8 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR* X. RR* ) ) )
13	8, 11, 12	mpbir2an 888	. . . . . . 7 |- <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) )
14	13	a1i 11	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ x = (/) ) -> <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) )
15		simplr 733	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x = ( a (,) b ) )
16	15	supeq1d 7480	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = sup ( ( a (,) b ) , RR* , `' < ) )
17		simplll 736	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a e. RR* )
18		simpllr 737	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> b e. RR* )
19		simpr 449	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> -. x = (/) )
20	19	neneqad 2680	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x =/= (/) )
21	15, 20	eqnetrrd 2627	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a (,) b ) =/= (/) )
22		df-ioo 10951	. . . . . . . . . . 11 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
23		idd 23	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w < b ) )
24		xrltle 10773	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w <_ b ) )
25		idd 23	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a < w ) )
26		xrltle 10773	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a <_ w ) )
27	22, 23, 24, 25, 26	ixxlb 10969	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
28	17, 18, 21, 27	syl3anc 1185	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
29	16, 28	eqtrd 2474	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = a )
30	15	supeq1d 7480	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = sup ( ( a (,) b ) , RR* , < ) )
31	22, 23, 24, 25, 26	ixxub 10968	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
32	17, 18, 21, 31	syl3anc 1185	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
33	30, 32	eqtrd 2474	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = b )
34	29, 33	opeq12d 4016	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. = <. a , b >. )
35		ioon0 10973	. . . . . . . . . . . 12 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
36	35	ad2antrr 708	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
37	21, 36	mpbid 203	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a < b )
38		xrltle 10773	. . . . . . . . . . 11 |- ( ( a e. RR* /\ b e. RR* ) -> ( a < b -> a <_ b ) )
39	38	ad2antrr 708	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a < b -> a <_ b ) )
40	37, 39	mpd 15	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a <_ b )
41		df-br 4238	. . . . . . . . 9 |- ( a <_ b <-> <. a , b >. e. <_ )
42	40, 41	sylib 190	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. <_ )
43		opelxpi 4939	. . . . . . . . 9 |- ( ( a e. RR* /\ b e. RR* ) -> <. a , b >. e. ( RR* X. RR* ) )
44	43	ad2antrr 708	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( RR* X. RR* ) )
45		elin 3516	. . . . . . . 8 |- ( <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. a , b >. e. <_ /\ <. a , b >. e. ( RR* X. RR* ) ) )
46	42, 44, 45	sylanbrc 647	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) )
47	34, 46	eqeltrd 2516	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. e. ( <_ i^i ( RR* X. RR* ) ) )
48	14, 47	ifclda 3790	. . . . 5 |- ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
49	48	ex 425	. . . 4 |- ( ( a e. RR* /\ b e. RR* ) -> ( x = ( a (,) b ) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) ) )
50	49	rexlimivv 2841	. . 3 |- ( E. a e. RR* E. b e. RR* x = ( a (,) b ) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
51	5, 50	sylbi 189	. 2 |- ( x e. ran (,) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
52	1, 51	fmpti 5921	1 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1727   =/= wne 2605   E.wrex 2712   i^i cin 3305   (/)c0 3613   ifcif 3763   ~Pcpw 3823   <.cop 3841   class class class wbr 4237   e. cmpt 4291   X. cxp 4905   `'ccnv 4906   ran crn 4908   Fn wfn 5478   -->wf 5479  (class class class)co 6110   supcsup 7474   RRcr 9020   0cc0 9021   RR*cxr 9150   < clt 9151   <_ cle 9152   (,)cioo 10947

This theorem is referenced by:  ioorcl  19500  uniioombllem2  19506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-q 10606  df-ioo 10951