Theorem iooval2 10037

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 10030	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2		elioore 10034	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. RR )
3	2	ssriv 3197	. . . . 5 |- ( A (,) B ) C_ RR
4	1, 3	eqsstrrdi 3246	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> { x e. RR* | ( A < x /\ x < B ) } C_ RR )
5		df-ss 3179	. . . 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 3446	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. ( RR* i^i RR ) | ( A < x /\ x < B ) }
8		ressxr 8116	. . . . . 6 |- RR C_ RR*
9		sseqin2 3392	. . . . . 6 |- ( RR C_ RR* <-> ( RR* i^i RR ) = RR )
10	8, 9	mpbi 145	. . . . 5 |- ( RR* i^i RR ) = RR
11		rabeq 2764	. . . . 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 2226	. . 3 |- ( { x e. RR* | ( A < x /\ x < B ) } i^i RR ) = { x e. RR | ( A < x /\ x < B ) }
14	6, 13	eqtr3di 2253	. 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 2238	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR | ( A < x /\ x < B ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {crab 2488   i^i cin 3165   C_ wss 3166   class class class wbr 4044  (class class class)co 5944   RRcr 7924   RR*cxr 8106   < clt 8107   (,)cioo 10010

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-ioo 10014

This theorem is referenced by:  elioo2  10043  ioomax  10070  ioopos  10072  dfioo2  10096