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Theorem rlimdmafv 37643
Description: Two ways to express that a function has a limit, analogous to rlimdm 13525. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
Hypotheses
Ref Expression
rlimdmafv.1  |-  ( ph  ->  F : A --> CC )
rlimdmafv.2  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
Assertion
Ref Expression
rlimdmafv  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r ''' F ) ) )

Proof of Theorem rlimdmafv
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldmg 5021 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 243 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 461 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 rlimrel 13467 . . . . . . . . . . . 12  |-  Rel  ~~> r
54brrelexi 4866 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  F  e.  _V )
65adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  _V )
7 vex 3064 . . . . . . . . . . 11  |-  x  e. 
_V
87a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  x  e.  _V )
9 breldmg 5031 . . . . . . . . . 10  |-  ( ( F  e.  _V  /\  x  e.  _V  /\  F  ~~> r  x )  ->  F  e.  dom  ~~> r  )
106, 8, 3, 9syl3anc 1232 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  dom 
~~> r  )
11 breq2 4401 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
1211biimprd 225 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  x  ->  F  ~~> r  y ) )
1312spimev 2039 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  E. y  F 
~~> r  y )
1413adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  E. y  F 
~~> r  y )
15 rlimdmafv.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> CC )
1615adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  ~~> r  x )  ->  F : A
--> CC )
1716adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F : A --> CC )
18 rlimdmafv.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
1918adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  ~~> r  x )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2019adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
21 simprl 758 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  y )
22 simprr 760 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  z )
2317, 20, 21, 22rlimuni 13524 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  y  =  z )
2423ex 434 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( ( F 
~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
2524alrimivv 1743 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  A. y A. z ( ( F  ~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
26 breq2 4401 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F 
~~> r  y  <->  F  ~~> r  z ) )
2726eu4 2292 . . . . . . . . . 10  |-  ( E! y  F  ~~> r  y  <-> 
( E. y  F  ~~> r  y  /\  A. y A. z ( ( F  ~~> r  y  /\  F 
~~> r  z )  -> 
y  =  z ) ) )
2814, 25, 27sylanbrc 664 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  E! y  F 
~~> r  y )
29 dfdfat2 37597 . . . . . . . . 9  |-  (  ~~> r defAt  F  <->  ( F  e.  dom  ~~> r  /\  E! y  F  ~~> r  y ) )
3010, 28, 29sylanbrc 664 . . . . . . . 8  |-  ( (
ph  /\  F  ~~> r  x )  ->  ~~> r defAt  F )
31 afvfundmfveq 37604 . . . . . . . 8  |-  (  ~~> r defAt  F  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F
) )
3230, 31syl 17 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F ) )
33 df-fv 5579 . . . . . . . 8  |-  (  ~~> r  `  F )  =  ( iota w F  ~~> r  w )
3415adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F : A --> CC )
3518adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
36 simprr 760 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  w )
37 simprl 758 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  x )
3834, 35, 36, 37rlimuni 13524 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  w  =  x )
3938expr 615 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  ->  w  =  x ) )
40 breq2 4401 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  ( F 
~~> r  w  <->  F  ~~> r  x ) )
413, 40syl5ibrcom 224 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( w  =  x  ->  F  ~~> r  w ) )
4239, 41impbid 192 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  <->  w  =  x
) )
4342adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  w  <->  w  =  x ) )
4443iota5 5555 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota w F  ~~> r  w )  =  x )
457, 44mpan2 671 . . . . . . . 8  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota w F  ~~> r  w )  =  x )
4633, 45syl5eq 2457 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
4732, 46eqtrd 2445 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  x )
483, 47breqtrrd 4423 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r ''' F ) )
4948ex 434 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
5049exlimdv 1747 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
512, 50syl5 32 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r ''' F ) ) )
524releldmi 5062 . 2  |-  ( F  ~~> r  (  ~~> r ''' F )  ->  F  e.  dom  ~~> r  )
5351, 52impbid1 205 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r ''' F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369   A.wal 1405    = wceq 1407   E.wex 1635    e. wcel 1844   E!weu 2240   _Vcvv 3061   class class class wbr 4397   dom cdm 4825   iotacio 5533   -->wf 5567   ` cfv 5571   supcsup 7936   CCcc 9522   +oocpnf 9657   RR*cxr 9659    < clt 9660    ~~> r crli 13459   defAt wdfat 37579  '''cafv 37580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-seq 12154  df-exp 12213  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-rlim 13463  df-dfat 37582  df-afv 37583
This theorem is referenced by: (None)
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