Theorem iooval2 9842

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 9835	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2		elioore 9839	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. RR )
3	2	ssriv 3141	. . . . 5 |- ( A (,) B ) C_ RR
4	1, 3	eqsstrrdi 3190	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } C_ RR )
5		df-ss 3124	. . . 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 3390	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) }
8		ressxr 7933	. . . . . 6 |- RR C_ RR*
9		sseqin2 3336	. . . . . 6 |- ( RR C_ RR* <-> ( RR* i^i RR ) = RR )
10	8, 9	mpbi 144	. . . . 5 |- ( RR* i^i RR ) = RR
11		rabeq 2713	. . . . 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 2185	. . 3 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR | ( A < x /\ x < B ) }
14	6, 13	eqtr3di 2212	. 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 2197	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   e. wcel 2135   {crab 2446   i^i cin 3110   C_ wss 3111   class class class wbr 3976  (class class class)co 5836   RRcr 7743   RR*cxr 7923   < clt 7924   (,)cioo 9815

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-po 4268  df-iso 4269  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-ioo 9819

This theorem is referenced by:  elioo2  9848  ioomax  9875  ioopos  9877  dfioo2  9901