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Theorem rlimdm 12300
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 5024 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 233 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 448 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 df-fv 5421 . . . . . . 7  |-  (  ~~> r  `  F )  =  ( iota y F  ~~> r  y )
5 vex 2919 . . . . . . . 8  |-  x  e. 
_V
6 rlimuni.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : A --> CC )
76adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F : A --> CC )
8 rlimuni.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
98adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
10 simprr 734 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  y )
11 simprl 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  x )
127, 9, 10, 11rlimuni 12299 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  y  =  x )
1312expr 599 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  ->  y  =  x ) )
14 breq2 4176 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
153, 14syl5ibrcom 214 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( y  =  x  ->  F  ~~> r  y ) )
1613, 15impbid 184 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  <->  y  =  x ) )
1716adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  y  <->  y  =  x ) )
1817iota5 5397 . . . . . . . 8  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota y F  ~~> r  y )  =  x )
195, 18mpan2 653 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota y F  ~~> r  y )  =  x )
204, 19syl5eq 2448 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
213, 20breqtrrd 4198 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r  `  F ) )
2221ex 424 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F ) ) )
2322exlimdv 1643 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F
) ) )
242, 23syl5 30 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r  `  F ) ) )
25 rlimrel 12242 . . 3  |-  Rel  ~~> r
2625releldmi 5065 . 2  |-  ( F  ~~> r  (  ~~> r  `  F )  ->  F  e.  dom  ~~> r  )
2724, 26impbid1 195 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   dom cdm 4837   iotacio 5375   -->wf 5409   ` cfv 5413   supcsup 7403   CCcc 8944    +oocpnf 9073   RR*cxr 9075    < clt 9076    ~~> r crli 12234
This theorem is referenced by:  caucvgrlem2  12423  caucvg  12427  dchrisum0lem3  21166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-rlim 12238
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