Theorem ioorfOLD 22609

Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorf 22604 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem ioorfOLD

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

Step	Hyp	Ref	Expression
1		ioorfOLD.1	. 2 |- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) )
2		ioof 11757	. . . 4 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5739	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4		ovelrn 6464	. . . 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 10721	. . . . . . . . 9 |- 0 <_ 0
7		df-br 4396	. . . . . . . . 9 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
8	6, 7	mpbi 213	. . . . . . . 8 |- <. 0 , 0 >. e. <_
9		0xr 9705	. . . . . . . . 9 |- 0 e. RR*
10		opelxpi 4871	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> <. 0 , 0 >. e. ( RR* X. RR* ) )
11	9, 9, 10	mp2an 686	. . . . . . . 8 |- <. 0 , 0 >. e. ( RR* X. RR* )
12		elin 3608	. . . . . . . 8 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR* X. RR* ) ) )
13	8, 11, 12	mpbir2an 934	. . . . . . 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 770	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x = ( a (,) b ) )
16	15	supeq1d 7978	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = sup ( ( a (,) b ) , RR* , `' < ) )
17		simplll 776	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a e. RR* )
18		simpllr 777	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> b e. RR* )
19		simpr 468	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> -. x = (/) )
20	19	neqned 2650	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x =/= (/) )
21	15, 20	eqnetrrd 2711	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a (,) b ) =/= (/) )
22		df-ioo 11664	. . . . . . . . . . 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 11471	. . . . . . . . . . 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 11471	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a <_ w ) )
27	22, 23, 24, 25, 26	ixxlbOLD 11683	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
28	17, 18, 21, 27	syl3anc 1292	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
29	16, 28	eqtrd 2505	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = a )
30	15	supeq1d 7978	. . . . . . . . 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 11681	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
32	17, 18, 21, 31	syl3anc 1292	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
33	30, 32	eqtrd 2505	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = b )
34	29, 33	opeq12d 4166	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. = <. a , b >. )
35		ioon0 11687	. . . . . . . . . . . 12 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
36	35	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
37	21, 36	mpbid 215	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a < b )
38		xrltle 11471	. . . . . . . . . . 11 |- ( ( a e. RR* /\ b e. RR* ) -> ( a < b -> a <_ b ) )
39	38	ad2antrr 740	. . . . . . . . . 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 4396	. . . . . . . . 9 |- ( a <_ b <-> <. a , b >. e. <_ )
42	40, 41	sylib 201	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. <_ )
43		opelxpi 4871	. . . . . . . . 9 |- ( ( a e. RR* /\ b e. RR* ) -> <. a , b >. e. ( RR* X. RR* ) )
44	43	ad2antrr 740	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( RR* X. RR* ) )
45	42, 44	elind 3609	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) )
46	34, 45	eqeltrd 2549	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. e. ( <_ i^i ( RR* X. RR* ) ) )
47	14, 46	ifclda 3904	. . . . 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* ) ) )
48	47	ex 441	. . . 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* ) ) ) )
49	48	rexlimivv 2876	. . 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* ) ) )
50	5, 49	sylbi 200	. 2 |- ( x e. ran (,) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
51	1, 50	fmpti 6060	1 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   i^i cin 3389   (/)c0 3722   ifcif 3872   ~Pcpw 3942   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   ran crn 4840   Fn wfn 5584   -->wf 5585  (class class class)co 6308   supcsup 7972   RRcr 9556   0cc0 9557   RR*cxr 9692   < clt 9693   <_ cle 9694   (,)cioo 11660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-ioo 11664

This theorem is referenced by:  ioorclOLD  22613  uniioombllem2OLD  22620