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Theorem rlimdm 12041
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 4890 . . . 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 df-fv 5279 . . . . . . 7  |-  (  ~~> r  `  F )  =  ( iota y F  ~~> r  y )
5 vex 2804 . . . . . . . 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 12040 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  y ) )  ->  y  =  x )
1312expr 598 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  y  ->  y  =  x ) )
14 breq2 4043 . . . . . . . . . . . 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 5255 . . . . . . . 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 2340 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
213, 20breqtrrd 4065 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r  `  F ) )
2221ex 423 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r  `  F ) ) )
2322exlimdv 1626 . . 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 11983 . . 3  |-  Rel  ~~> r
2625releldmi 4931 . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   dom cdm 4705   iotacio 5233   -->wf 5267   ` cfv 5271   supcsup 7209   CCcc 8751    +oocpnf 8880   RR*cxr 8882    < clt 8883    ~~> r crli 11975
This theorem is referenced by:  caucvgrlem2  12163  caucvg  12167  dchrisum0lem3  20684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-rlim 11979
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