Theorem iooval2 9851

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 9844	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2		elioore 9848	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. RR )
3	2	ssriv 3146	. . . . 5 |- ( A (,) B ) C_ RR
4	1, 3	eqsstrrdi 3195	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } C_ RR )
5		df-ss 3129	. . . 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 121	. . 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 3395	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) }
8		ressxr 7942	. . . . . 6 |- RR C_ RR*
9		sseqin2 3341	. . . . . 6 |- ( RR C_ RR* <-> ( RR* i^i RR ) = RR )
10	8, 9	mpbi 144	. . . . 5 |- ( RR* i^i RR ) = RR
11		rabeq 2718	. . . . 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 2186	. . 3 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR | ( A < x /\ x < B ) }
14	6, 13	eqtr3di 2214	. 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 2198	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {crab 2448   i^i cin 3115   C_ wss 3116   class class class wbr 3982  (class class class)co 5842   RRcr 7752   RR*cxr 7932   < clt 7933   (,)cioo 9824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-ioo 9828

This theorem is referenced by:  elioo2  9857  ioomax  9884  ioopos  9886  dfioo2  9910