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Theorem rlimdm 12021
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 4873 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 232 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 447 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 fv4 6282 . . . . . . 7  |-  (  ~~> r  `  F )  =  ( iota y F  ~~> r  y )
5 vex 2792 . . . . . . . 8  |-  x  e. 
_V
6 rlimuni.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : A --> CC )
76adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F : A --> CC )
8 rlimuni.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
98adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
10 simprr 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  y )
11 simprl 732 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  F  ~~> r  x )
127, 9, 10, 11rlimuni 12020 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  y  =  x )
1312expr 598 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  ->  y  =  x ) )
14 breq2 4028 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
153, 14syl5ibrcom 213 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( y  =  x  ->  F  ~~> r  y ) )
1613, 15impbid 183 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  <->  y  =  x ) )
1716adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  y  <->  y  =  x ) )
1817iota5 6273 . . . . . . . 8  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota y F  ~~> r  y )  =  x )
195, 18mpan2 652 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota y F  ~~> r  y )  =  x )
204, 19syl5eq 2328 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
213, 20breqtrrd 4050 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r  `  F ) )
2221ex 423 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F ) ) )
2322exlimdv 1665 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F
) ) )
242, 23syl5 28 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r  `  F ) ) )
25 rlimrel 11963 . . 3  |-  Rel  ~~> r
2625releldmi 4914 . 2  |-  ( F  ~~> r  (  ~~> r  `  F )  ->  F  e.  dom  ~~> r  )
2724, 26impbid1 194 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r  `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   _Vcvv 2789   class class class wbr 4024    dom cdm 4688   -->wf 5217   ` cfv 5221   iotacio 6251   supcsup 7189   CCcc 8731    +oocpnf 8860   RR*cxr 8862    < clt 8863    ~~> r crli 11955
This theorem is referenced by:  caucvgrlem2  12143  caucvg  12147  dchrisum0lem3  20664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-pm 6771  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-rlim 11959
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