Theorem ioorf 23341

Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)

Hypothesis

Ref	Expression
ioorf.1	|- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <.inf ( 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 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) )
2		ioof 12271	. . . 4 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 6045	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4		ovelrn 6810	. . . 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 11110	. . . . . . . . 9 |- 0 <_ 0
7		df-br 4654	. . . . . . . . 9 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
8	6, 7	mpbi 220	. . . . . . . 8 |- <. 0 , 0 >. e. <_
9		0xr 10086	. . . . . . . . 9 |- 0 e. RR*
10		opelxpi 5148	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> <. 0 , 0 >. e. ( RR* X. RR* ) )
11	9, 9, 10	mp2an 708	. . . . . . . 8 |- <. 0 , 0 >. e. ( RR* X. RR* )
12		elin 3796	. . . . . . . 8 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR* X. RR* ) ) )
13	8, 11, 12	mpbir2an 955	. . . . . . 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 792	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x = ( a (,) b ) )
16	15	infeq1d 8383	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> inf ( x , RR* , < ) = inf ( ( a (,) b ) , RR* , < ) )
17		simplll 798	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a e. RR* )
18		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> b e. RR* )
19		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> -. x = (/) )
20	19	neqned 2801	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x =/= (/) )
21	15, 20	eqnetrrd 2862	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a (,) b ) =/= (/) )
22		df-ioo 12179	. . . . . . . . . . 11 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
23		idd 24	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w < b ) )
24		xrltle 11982	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w <_ b ) )
25		idd 24	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a < w ) )
26		xrltle 11982	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a <_ w ) )
27	22, 23, 24, 25, 26	ixxlb 12197	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> inf ( ( a (,) b ) , RR* , < ) = a )
28	17, 18, 21, 27	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> inf ( ( a (,) b ) , RR* , < ) = a )
29	16, 28	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> inf ( x , RR* , < ) = a )
30	15	supeq1d 8352	. . . . . . . . 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 12196	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
32	17, 18, 21, 31	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
33	30, 32	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = b )
34	29, 33	opeq12d 4410	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. = <. a , b >. )
35		ioon0 12201	. . . . . . . . . . . 12 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
36	35	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
37	21, 36	mpbid 222	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a < b )
38		xrltle 11982	. . . . . . . . . . 11 |- ( ( a e. RR* /\ b e. RR* ) -> ( a < b -> a <_ b ) )
39	38	ad2antrr 762	. . . . . . . . . 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 4654	. . . . . . . . 9 |- ( a <_ b <-> <. a , b >. e. <_ )
42	40, 41	sylib 208	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. <_ )
43		opelxpi 5148	. . . . . . . . 9 |- ( ( a e. RR* /\ b e. RR* ) -> <. a , b >. e. ( RR* X. RR* ) )
44	43	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( RR* X. RR* ) )
45	42, 44	elind 3798	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) )
46	34, 45	eqeltrd 2701	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. e. ( <_ i^i ( RR* X. RR* ) ) )
47	14, 46	ifclda 4120	. . . . 5 |- ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) -> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
48	47	ex 450	. . . 4 |- ( ( a e. RR* /\ b e. RR* ) -> ( x = ( a (,) b ) -> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) ) )
49	48	rexlimivv 3036	. . 3 |- ( E. a e. RR* E. b e. RR* x = ( a (,) b ) -> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
50	5, 49	sylbi 207	. 2 |- ( x e. ran (,) -> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
51	1, 50	fmpti 6383	1 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   (/)c0 3915   ifcif 4086   ~Pcpw 4158   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179

This theorem is referenced by:  ioorcl  23345  uniioombllem2  23351