Theorem elicores 39760

Description: Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Assertion

Ref	Expression
elicores	|- ( A e. ran ( [,) |` ( RR X. RR ) ) <-> E. x e. RR E. y e. RR A = ( x [,) y ) )

Distinct variable group:   x, A, y

Proof of Theorem elicores

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ico 12181	. . . . . 6 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	reseq1i 5392	. . . . 5 |- ( [,) |` ( RR X. RR ) ) = ( ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } ) |` ( RR X. RR ) )
3		ressxr 10083	. . . . . 6 |- RR C_ RR*
4		resmpt2 6758	. . . . . 6 |- ( ( RR C_ RR* /\ RR C_ RR* ) -> ( ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } ) |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } ) )
5	3, 3, 4	mp2an 708	. . . . 5 |- ( ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } ) |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6	2, 5	eqtri 2644	. . . 4 |- ( [,) |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } )
7	6	rneqi 5352	. . 3 |- ran ( [,) |` ( RR X. RR ) ) = ran ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } )
8	7	eleq2i 2693	. 2 |- ( A e. ran ( [,) |` ( RR X. RR ) ) <-> A e. ran ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } ) )
9		eqid 2622	. . 3 |- ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } ) = ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } )
10		xrex 11829	. . . 4 |- RR* e. _V
11	10	rabex 4813	. . 3 |- { z e. RR* | ( x <_ z /\ z < y ) } e. _V
12	9, 11	elrnmpt2 6773	. 2 |- ( A e. ran ( x e. RR , y e. RR |-> { z e. RR* | ( x <_ z /\ z < y ) } ) <-> E. x e. RR E. y e. RR A = { z e. RR* | ( x <_ z /\ z < y ) } )
13	3	sseli 3599	. . . . . . . 8 |- ( x e. RR -> x e. RR* )
14	13	adantr 481	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> x e. RR* )
15	3	sseli 3599	. . . . . . . 8 |- ( y e. RR -> y e. RR* )
16	15	adantl 482	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> y e. RR* )
17		icoval 12213	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x [,) y ) = { z e. RR* | ( x <_ z /\ z < y ) } )
18	14, 16, 17	syl2anc 693	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x [,) y ) = { z e. RR* | ( x <_ z /\ z < y ) } )
19	18	eqcomd 2628	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> { z e. RR* | ( x <_ z /\ z < y ) } = ( x [,) y ) )
20	19	eqeq2d 2632	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( A = { z e. RR* | ( x <_ z /\ z < y ) } <-> A = ( x [,) y ) ) )
21	20	rexbidva 3049	. . 3 |- ( x e. RR -> ( E. y e. RR A = { z e. RR* | ( x <_ z /\ z < y ) } <-> E. y e. RR A = ( x [,) y ) ) )
22	21	rexbiia 3040	. 2 |- ( E. x e. RR E. y e. RR A = { z e. RR* | ( x <_ z /\ z < y ) } <-> E. x e. RR E. y e. RR A = ( x [,) y ) )
23	8, 12, 22	3bitri 286	1 |- ( A e. ran ( [,) |` ( RR X. RR ) ) <-> E. x e. RR E. y e. RR A = ( x [,) y ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078  df-ico 12181

This theorem is referenced by:  icoresmbl  40757