Theorem ioorf 18928

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 10741	. . . 4 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5389	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4		ovelrn 5996	. . . 4 |- ( (,) Fn ( RR* X. RR* ) -> ( x e. ran (,) <-> E. a e. RR* E. b e. RR* x = ( a (,) b ) ) )
5	2, 3, 4	mp2b 9	. . 3 |- ( x e. ran (,) <-> E. a e. RR* E. b e. RR* x = ( a (,) b ) )
6		0le0 9827	. . . . . . . . 9 |- 0 <_ 0
7		df-br 4024	. . . . . . . . 9 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
8	6, 7	mpbi 199	. . . . . . . 8 |- <. 0 , 0 >. e. <_
9		0xr 8878	. . . . . . . . 9 |- 0 e. RR*
10		opelxpi 4721	. . . . . . . . 9 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> <. 0 , 0 >. e. ( RR* X. RR* ) )
11	9, 9, 10	mp2an 653	. . . . . . . 8 |- <. 0 , 0 >. e. ( RR* X. RR* )
12		elin 3358	. . . . . . . 8 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR* X. RR* ) ) )
13	8, 11, 12	mpbir2an 886	. . . . . . 7 |- <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) )
14	13	a1i 10	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ x = (/) ) -> <. 0 , 0 >. e. ( <_ i^i ( RR* X. RR* ) ) )
15		simplr 731	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x = ( a (,) b ) )
16	15	supeq1d 7199	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = sup ( ( a (,) b ) , RR* , `' < ) )
17		simplll 734	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a e. RR* )
18		simpllr 735	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> b e. RR* )
19		simpr 447	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> -. x = (/) )
20		df-ne 2448	. . . . . . . . . . . 12 |- ( x =/= (/) <-> -. x = (/) )
21	19, 20	sylibr 203	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> x =/= (/) )
22	15, 21	eqnetrrd 2466	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a (,) b ) =/= (/) )
23		df-ioo 10660	. . . . . . . . . . 11 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
24		idd 21	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w < b ) )
25		xrltle 10483	. . . . . . . . . . 11 |- ( ( w e. RR* /\ b e. RR* ) -> ( w < b -> w <_ b ) )
26		idd 21	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a < w ) )
27		xrltle 10483	. . . . . . . . . . 11 |- ( ( a e. RR* /\ w e. RR* ) -> ( a < w -> a <_ w ) )
28	23, 24, 25, 26, 27	ixxlb 10678	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
29	17, 18, 22, 28	syl3anc 1182	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , `' < ) = a )
30	16, 29	eqtrd 2315	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , `' < ) = a )
31	15	supeq1d 7199	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = sup ( ( a (,) b ) , RR* , < ) )
32	23, 24, 25, 26, 27	ixxub 10677	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* /\ ( a (,) b ) =/= (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
33	17, 18, 22, 32	syl3anc 1182	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( ( a (,) b ) , RR* , < ) = b )
34	31, 33	eqtrd 2315	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> sup ( x , RR* , < ) = b )
35	30, 34	opeq12d 3804	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. = <. a , b >. )
36		ioon0 10682	. . . . . . . . . . . 12 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
37	36	ad2antrr 706	. . . . . . . . . . 11 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( ( a (,) b ) =/= (/) <-> a < b ) )
38	22, 37	mpbid 201	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a < b )
39		xrltle 10483	. . . . . . . . . . 11 |- ( ( a e. RR* /\ b e. RR* ) -> ( a < b -> a <_ b ) )
40	39	ad2antrr 706	. . . . . . . . . 10 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> ( a < b -> a <_ b ) )
41	38, 40	mpd 14	. . . . . . . . 9 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> a <_ b )
42		df-br 4024	. . . . . . . . 9 |- ( a <_ b <-> <. a , b >. e. <_ )
43	41, 42	sylib 188	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. <_ )
44		opelxpi 4721	. . . . . . . . 9 |- ( ( a e. RR* /\ b e. RR* ) -> <. a , b >. e. ( RR* X. RR* ) )
45	44	ad2antrr 706	. . . . . . . 8 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( RR* X. RR* ) )
46		elin 3358	. . . . . . . 8 |- ( <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) <-> ( <. a , b >. e. <_ /\ <. a , b >. e. ( RR* X. RR* ) ) )
47	43, 45, 46	sylanbrc 645	. . . . . . 7 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. a , b >. e. ( <_ i^i ( RR* X. RR* ) ) )
48	35, 47	eqeltrd 2357	. . . . . 6 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ x = ( a (,) b ) ) /\ -. x = (/) ) -> <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. e. ( <_ i^i ( RR* X. RR* ) ) )
49	14, 48	ifclda 3592	. . . . 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* ) ) )
50	49	ex 423	. . . 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* ) ) ) )
51	50	rexlimivv 2672	. . 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* ) ) )
52	5, 51	sylbi 187	. 2 |- ( x e. ran (,) -> if ( x = (/) , <. 0 , 0 >. , <. sup ( x , RR* , `' < ) , sup ( x , RR* , < ) >. ) e. ( <_ i^i ( RR* X. RR* ) ) )
53	1, 52	fmpti 5683	1 |- F : ran (,) --> ( <_ i^i ( RR* X. RR* ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1623   e. wcel 1684   =/= wne 2446   E.wrex 2544   i^i cin 3151   (/)c0 3455   ifcif 3565   ~Pcpw 3625   <.cop 3643   class class class wbr 4023   e. cmpt 4077   X. cxp 4687   `'ccnv 4688   ran crn 4690   Fn wfn 5250   -->wf 5251  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   RR*cxr 8866   < clt 8867   <_ cle 8868   (,)cioo 10656

This theorem is referenced by:  ioorcl  18932  uniioombllem2  18938

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660