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Theorem rlimdm 13593
Description: Two ways to express that a function has a limit. (The expression  (  ~~> r  `  F ) is sometimes useful as a shorthand for "the unique limit of the function  F"). (Contributed by Mario Carneiro, 8-May-2016.)
Hypotheses
Ref Expression
rlimuni.1  |-  ( ph  ->  F : A --> CC )
rlimuni.2  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
Assertion
Ref Expression
rlimdm  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )

Proof of Theorem rlimdm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldmg 5050 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 244 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 462 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 df-fv 5609 . . . . . . 7  |-  (  ~~> r  `  F )  =  ( iota y F  ~~> r  y )
5 vex 3090 . . . . . . . 8  |-  x  e. 
_V
6 rlimuni.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : A --> CC )
76adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F : A --> CC )
8 rlimuni.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
98adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
10 simprr 764 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  y )
11 simprl 762 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  x )
127, 9, 10, 11rlimuni 13592 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  y  =  x )
1312expr 618 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  ->  y  =  x ) )
14 breq2 4430 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
153, 14syl5ibrcom 225 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( y  =  x  ->  F  ~~> r  y ) )
1613, 15impbid 193 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  <->  y  =  x ) )
1716adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  y  <->  y  =  x ) )
1817iota5 5585 . . . . . . . 8  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota y F  ~~> r  y )  =  x )
195, 18mpan2 675 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota y F  ~~> r  y )  =  x )
204, 19syl5eq 2482 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
213, 20breqtrrd 4452 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r  `  F ) )
2221ex 435 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F ) ) )
2322exlimdv 1771 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F
) ) )
242, 23syl5 33 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r  `  F ) ) )
25 rlimrel 13535 . . 3  |-  Rel  ~~> r
2625releldmi 5091 . 2  |-  ( F  ~~> r  (  ~~> r  `  F )  ->  F  e.  dom  ~~> r  )
2724, 26impbid1 206 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   _Vcvv 3087   class class class wbr 4426   dom cdm 4854   iotacio 5563   -->wf 5597   ` cfv 5601   supcsup 7960   CCcc 9536   +oocpnf 9671   RR*cxr 9673    < clt 9674    ~~> r crli 13527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-rlim 13531
This theorem is referenced by:  caucvgrlem2  13718  caucvg  13723  dchrisum0lem3  24220
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