Theorem iooval2 9947

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
iooval2	|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Distinct variable groups:   x, A   x, B

Proof of Theorem iooval2

Step	Hyp	Ref	Expression
1		iooval 9940	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2		elioore 9944	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. RR )
3	2	ssriv 3174	. . . . 5 |- ( A (,) B ) C_ RR
4	1, 3	eqsstrrdi 3223	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } C_ RR )
5		df-ss 3157	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } C_ RR <-> ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR* | ( A < x /\ x < B ) } )
6	4, 5	sylib 122	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR* | ( A < x /\ x < B ) } )
7		inrab2 3423	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) }
8		ressxr 8032	. . . . . 6 |- RR C_ RR*
9		sseqin2 3369	. . . . . 6 |- ( RR C_ RR* <-> ( RR* i^i RR ) = RR )
10	8, 9	mpbi 145	. . . . 5 |- ( RR* i^i RR ) = RR
11		rabeq 2744	. . . . 5 |- ( ( RR* i^i RR ) = RR -> { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) } = { x e. RR | ( A < x /\ x < B ) } )
12	10, 11	ax-mp 5	. . . 4 |- { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) } = { x e. RR | ( A < x /\ x < B ) }
13	7, 12	eqtri 2210	. . 3 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR | ( A < x /\ x < B ) }
14	6, 13	eqtr3di 2237	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } = { x e. RR | ( A < x /\ x < B ) } )
15	1, 14	eqtrd 2222	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   {crab 2472   i^i cin 3143   C_ wss 3144   class class class wbr 4018  (class class class)co 5897   RRcr 7841   RR*cxr 8022   < clt 8023   (,)cioo 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-ioo 9924

This theorem is referenced by:  elioo2  9953  ioomax  9980  ioopos  9982  dfioo2  10006