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Theorem leordtvallem1 18947
Description: Lemma for leordtval 18950. (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
StepHypRef Expression
1 leordtval.1 . 2  |-  A  =  ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )
2 iocssxr 11491 . . . . . 6  |-  ( x (,] +oo )  C_  RR*
3 dfss1 3664 . . . . . 6  |-  ( ( x (,] +oo )  C_ 
RR* 
<->  ( RR*  i^i  (
x (,] +oo )
)  =  ( x (,] +oo ) )
42, 3mpbi 208 . . . . 5  |-  ( RR*  i^i  ( x (,] +oo ) )  =  ( x (,] +oo )
5 simpl 457 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
6 pnfxr 11204 . . . . . . . 8  |- +oo  e.  RR*
7 elioc1 11454 . . . . . . . 8  |-  ( ( x  e.  RR*  /\ +oo  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
85, 6, 7sylancl 662 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
9 simpr 461 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
10 pnfge 11222 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  y  <_ +oo )
119, 10jccir 539 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR*  /\  y  <_ +oo ) )
1211biantrurd 508 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( (
y  e.  RR*  /\  y  <_ +oo )  /\  x  <  y ) ) )
13 3anan32 977 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  x  <  y  /\  y  <_ +oo )  <->  ( ( y  e.  RR*  /\  y  <_ +oo )  /\  x  <  y ) )
1412, 13syl6bbr 263 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  ( y  e.  RR*  /\  x  < 
y  /\  y  <_ +oo ) ) )
15 xrltnle 9555 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
168, 14, 153bitr2d 281 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  ( x (,] +oo )  <->  -.  y  <_  x ) )
1716rabbi2dva 3667 . . . . 5  |-  ( x  e.  RR*  ->  ( RR*  i^i  ( x (,] +oo ) )  =  {
y  e.  RR*  |  -.  y  <_  x } )
184, 17syl5eqr 2509 . . . 4  |-  ( x  e.  RR*  ->  ( x (,] +oo )  =  { y  e.  RR*  |  -.  y  <_  x } )
1918mpteq2ia 4483 . . 3  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
2019rneqi 5175 . 2  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
211, 20eqtri 2483 1  |-  A  =  ran  ( x  e. 
RR*  |->  { y  e. 
RR*  |  -.  y  <_  x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803    i^i cin 3436    C_ wss 3437   class class class wbr 4401    |-> cmpt 4459   ran crn 4950  (class class class)co 6201   +oocpnf 9527   RR*cxr 9529    < clt 9530    <_ cle 9531   (,]cioc 11413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-ioc 11417
This theorem is referenced by:  leordtval2  18949  leordtval  18950
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