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Theorem rlimdmafv 30221
Description: Two ways to express that a function has a limit, analogous to rlimdm 13131. (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 5133 . . . 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 13073 . . . . . . . . . . . 12  |-  Rel  ~~> r
54brrelexi 4977 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  F  e.  _V )
65adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  _V )
7 vex 3071 . . . . . . . . . . 11  |-  x  e. 
_V
87a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  x  e.  _V )
9 breldmg 5143 . . . . . . . . . 10  |-  ( ( F  e.  _V  /\  x  e.  _V  /\  F  ~~> r  x )  ->  F  e.  dom  ~~> r  )
106, 8, 3, 9syl3anc 1219 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  dom 
~~> r  )
11 breq2 4394 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
1211biimprd 223 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  x  ->  F  ~~> r  y ) )
1312spimev 1963 . . . . . . . . . . 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 13130 . . . . . . . . . . . 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 1687 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  A. y A. z ( ( F  ~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
26 breq2 4394 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F 
~~> r  y  <->  F  ~~> r  z ) )
2726eu4 2325 . . . . . . . . . 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 30175 . . . . . . . . 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 30182 . . . . . . . 8  |-  (  ~~> r defAt  F  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F
) )
3230, 31syl 16 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F ) )
33 df-fv 5524 . . . . . . . 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 13130 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  w  =  x )
3938expr 615 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  ->  w  =  x ) )
40 breq2 4394 . . . . . . . . . . . . 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 5499 . . . . . . . . 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 2504 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
4732, 46eqtrd 2492 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  x )
483, 47breqtrrd 4416 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r ''' F ) )
4948ex 434 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
5049exlimdv 1691 . . 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 5174 . 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 1368    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2260   _Vcvv 3068   class class class wbr 4390   dom cdm 4938   iotacio 5477   -->wf 5512   ` cfv 5516   supcsup 7791   CCcc 9381   +oocpnf 9516   RR*cxr 9518    < clt 9519    ~~> r crli 13065   defAt wdfat 30155  '''cafv 30156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-rlim 13069  df-dfat 30158  df-afv 30159
This theorem is referenced by: (None)
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