Theorem ioorf 19422

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 10962	. . . 4 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5554	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4		ovelrn 6185	. . . 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 10041	. . . . . . . . 9 |- 0 <_ 0
7		df-br 4177	. . . . . . . . 9 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
8	6, 7	mpbi 200	. . . . . . . 8 |- <. 0 , 0 >. e. <_
9		0xr 9091	. . . . . . . . 9 |- 0 e. RR*
10		opelxpi 4873	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> <. 0 , 0 >. e. ( RR* X. RR* ) )
11	9, 9, 10	mp2an 654	. . . . . . . 8 |- <. 0 , 0 >. e. ( RR* X. RR* )
12		elin 3494	. . . . . . . 8 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR* X. RR* ) ) )
13	8, 11, 12	mpbir2an 887	. . . . . . 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 732	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x = ( a (,) b ) )
16	15	supeq1d 7413	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = sup ( ( a (,) b ) , RR* , `' < ) )
17		simplll 735	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a e. RR* )
18		simpllr 736	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> b e. RR* )
19		simpr 448	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> -. x = (/) )
20	19	neneqad 2641	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x =/= (/) )
21	15, 20	eqnetrrd 2591	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a (,) b ) =/= (/) )
22		df-ioo 10880	. . . . . . . . . . 11 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
23		idd 22	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w < b ) )
24		xrltle 10702	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w <_ b ) )
25		idd 22	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a < w ) )
26		xrltle 10702	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a <_ w ) )
27	22, 23, 24, 25, 26	ixxlb 10898	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
28	17, 18, 21, 27	syl3anc 1184	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
29	16, 28	eqtrd 2440	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = a )
30	15	supeq1d 7413	. . . . . . . . 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 10897	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
32	17, 18, 21, 31	syl3anc 1184	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
33	30, 32	eqtrd 2440	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = b )
34	29, 33	opeq12d 3956	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. = <. a , b >. )
35		ioon0 10902	. . . . . . . . . . . 12 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
36	35	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
37	21, 36	mpbid 202	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a < b )
38		xrltle 10702	. . . . . . . . . . 11 |- ( ( a e. RR* /\ b e. RR* ) -> ( a < b -> a <_ b ) )
39	38	ad2antrr 707	. . . . . . . . . 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 4177	. . . . . . . . 9 |- ( a <_ b <-> <. a , b >. e. <_ )
42	40, 41	sylib 189	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. <_ )
43		opelxpi 4873	. . . . . . . . 9 |- ( ( a e. RR* /\ b e. RR* ) -> <. a , b >. e. ( RR* X. RR* ) )
44	43	ad2antrr 707	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( RR* X. RR* ) )
45		elin 3494	. . . . . . . 8 |- ( <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. a , b >. e. <_ /\ <. a , b >. e. ( RR* X. RR* ) ) )
46	42, 44, 45	sylanbrc 646	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) )
47	34, 46	eqeltrd 2482	. . . . . 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 3730	. . . . 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 424	. . . 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 2799	. . 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 188	. 2 |- ( x e. ran (,) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
52	1, 51	fmpti 5855	1 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2571   E.wrex 2671   i^i cin 3283   (/)c0 3592   ifcif 3703   ~Pcpw 3763   <.cop 3781   class class class wbr 4176   e. cmpt 4230   X. cxp 4839   `'ccnv 4840   ran crn 4842   Fn wfn 5412   -->wf 5413  (class class class)co 6044   supcsup 7407   RRcr 8949   0cc0 8950   RR*cxr 9079   < clt 9080   <_ cle 9081   (,)cioo 10876

This theorem is referenced by:  ioorcl  19426  uniioombllem2  19432

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-ioo 10880