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Theorem rlimdmafv 31729
Description: Two ways to express that a function has a limit, analogous to rlimdm 13333. (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 5196 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 241 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 461 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 rlimrel 13275 . . . . . . . . . . . 12  |-  Rel  ~~> r
54brrelexi 5039 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  F  e.  _V )
65adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  _V )
7 vex 3116 . . . . . . . . . . 11  |-  x  e. 
_V
87a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  x  e.  _V )
9 breldmg 5206 . . . . . . . . . 10  |-  ( ( F  e.  _V  /\  x  e.  _V  /\  F  ~~> r  x )  ->  F  e.  dom  ~~> r  )
106, 8, 3, 9syl3anc 1228 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  dom 
~~> r  )
11 breq2 4451 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
1211biimprd 223 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  x  ->  F  ~~> r  y ) )
1312spimev 1979 . . . . . . . . . . 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 755 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  y )
22 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  z )
2317, 20, 21, 22rlimuni 13332 . . . . . . . . . . . 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 1696 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  A. y A. z ( ( F  ~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
26 breq2 4451 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F 
~~> r  y  <->  F  ~~> r  z ) )
2726eu4 2340 . . . . . . . . . 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 31683 . . . . . . . . 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 31690 . . . . . . . 8  |-  (  ~~> r defAt  F  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F
) )
3230, 31syl 16 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F ) )
33 df-fv 5594 . . . . . . . 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 756 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  w )
37 simprl 755 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  x )
3834, 35, 36, 37rlimuni 13332 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  w  =  x )
3938expr 615 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  ->  w  =  x ) )
40 breq2 4451 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  ( F 
~~> r  w  <->  F  ~~> r  x ) )
413, 40syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( w  =  x  ->  F  ~~> r  w ) )
4239, 41impbid 191 . . . . . . . . . . 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 5569 . . . . . . . . 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 2520 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
4732, 46eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  x )
483, 47breqtrrd 4473 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r ''' F ) )
4948ex 434 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
5049exlimdv 1700 . . 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 5237 . 2  |-  ( F  ~~> r  (  ~~> r ''' F )  ->  F  e.  dom  ~~> r  )
5351, 52impbid1 203 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r ''' F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   _Vcvv 3113   class class class wbr 4447   dom cdm 4999   iotacio 5547   -->wf 5582   ` cfv 5586   supcsup 7896   CCcc 9486   +oocpnf 9621   RR*cxr 9623    < clt 9624    ~~> r crli 13267   defAt wdfat 31665  '''cafv 31666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-rlim 13271  df-dfat 31668  df-afv 31669
This theorem is referenced by: (None)
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