Theorem iooval2 9917

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 9910	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2		elioore 9914	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. RR )
3	2	ssriv 3161	. . . . 5 |- ( A (,) B ) C_ RR
4	1, 3	eqsstrrdi 3210	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } C_ RR )
5		df-ss 3144	. . . 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 3410	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) }
8		ressxr 8003	. . . . . 6 |- RR C_ RR*
9		sseqin2 3356	. . . . . 6 |- ( RR C_ RR* <-> ( RR* i^i RR ) = RR )
10	8, 9	mpbi 145	. . . . 5 |- ( RR* i^i RR ) = RR
11		rabeq 2731	. . . . 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 2198	. . 3 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR | ( A < x /\ x < B ) }
14	6, 13	eqtr3di 2225	. 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 2210	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {crab 2459   i^i cin 3130   C_ wss 3131   class class class wbr 4005  (class class class)co 5877   RRcr 7812   RR*cxr 7993   < clt 7994   (,)cioo 9890

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-ioo 9894

This theorem is referenced by:  elioo2  9923  ioomax  9950  ioopos  9952  dfioo2  9976