Theorem leordtvallem1 21014

Description: Lemma for leordtval 21017. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
leordtval.1	|- A = ran ( x e. RR* |-> ( x (,] +oo ) )

Assertion

Ref	Expression
leordtvallem1	|- A = ran ( x e. RR* |-> { y e. RR* | -. y <_ x } )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem leordtvallem1

Step	Hyp	Ref	Expression
1		leordtval.1	. 2 |- A = ran ( x e. RR* |-> ( x (,] +oo ) )
2		iocssxr 12257	. . . . . 6 |- ( x (,] +oo ) C_ RR*
3		sseqin2 3817	. . . . . 6 |- ( ( x (,] +oo ) C_ RR* <-> ( RR* i^i ( x (,] +oo ) ) = ( x (,] +oo ) )
4	2, 3	mpbi 220	. . . . 5 |- ( RR* i^i ( x (,] +oo ) ) = ( x (,] +oo )
5		simpl 473	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> x e. RR* )
6		pnfxr 10092	. . . . . . . 8 |- +oo e. RR*
7		elioc1 12217	. . . . . . . 8 |- ( ( x e. RR* /\ +oo e. RR* ) -> ( y e. ( x (,] +oo ) <-> ( y e. RR* /\ x < y /\ y <_ +oo ) ) )
8	5, 6, 7	sylancl 694	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( y e. ( x (,] +oo ) <-> ( y e. RR* /\ x < y /\ y <_ +oo ) ) )
9		simpr 477	. . . . . . . . . 10 |- ( ( x e. RR* /\ y e. RR* ) -> y e. RR* )
10		pnfge 11964	. . . . . . . . . 10 |- ( y e. RR* -> y <_ +oo )
11	9, 10	jccir 562	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> ( y e. RR* /\ y <_ +oo ) )
12	11	biantrurd 529	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> ( x < y <-> ( ( y e. RR* /\ y <_ +oo ) /\ x < y ) ) )
13		3anan32 1050	. . . . . . . 8 |- ( ( y e. RR* /\ x < y /\ y <_ +oo ) <-> ( ( y e. RR* /\ y <_ +oo ) /\ x < y ) )
14	12, 13	syl6bbr 278	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x < y <-> ( y e. RR* /\ x < y /\ y <_ +oo ) ) )
15		xrltnle 10105	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x < y <-> -. y <_ x ) )
16	8, 14, 15	3bitr2d 296	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> ( y e. ( x (,] +oo ) <-> -. y <_ x ) )
17	16	rabbi2dva 3821	. . . . 5 |- ( x e. RR* -> ( RR* i^i ( x (,] +oo ) ) = { y e. RR* | -. y <_ x } )
18	4, 17	syl5eqr 2670	. . . 4 |- ( x e. RR* -> ( x (,] +oo ) = { y e. RR* | -. y <_ x } )
19	18	mpteq2ia 4740	. . 3 |- ( x e. RR* |-> ( x (,] +oo ) ) = ( x e. RR* |-> { y e. RR* | -. y <_ x } )
20	19	rneqi 5352	. 2 |- ran ( x e. RR* |-> ( x (,] +oo ) ) = ran ( x e. RR* |-> { y e. RR* | -. y <_ x } )
21	1, 20	eqtri 2644	1 |- A = ran ( x e. RR* |-> { y e. RR* | -. y <_ x } )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115  (class class class)co 6650   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176

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-pow 4843  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioc 12180

This theorem is referenced by:  leordtval2  21016  leordtval  21017