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Theorem rlimdmafv 38673
Description: Two ways to express that a function has a limit, analogous to rlimdm 13608. (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 5029 . . . 4  |-  ( F  e.  dom  ~~> r  -> 
( F  e.  dom  ~~> r  <->  E. x  F  ~~> r  x ) )
21ibi 245 . . 3  |-  ( F  e.  dom  ~~> r  ->  E. x  F  ~~> r  x )
3 simpr 463 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  x )
4 rlimrel 13550 . . . . . . . . . . . 12  |-  Rel  ~~> r
54brrelexi 4874 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  F  e.  _V )
65adantl 468 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  _V )
7 vex 3047 . . . . . . . . . . 11  |-  x  e. 
_V
87a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  x  e.  _V )
9 breldmg 5039 . . . . . . . . . 10  |-  ( ( F  e.  _V  /\  x  e.  _V  /\  F  ~~> r  x )  ->  F  e.  dom  ~~> r  )
106, 8, 3, 9syl3anc 1267 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  e.  dom 
~~> r  )
11 breq2 4405 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( F 
~~> r  y  <->  F  ~~> r  x ) )
1211biimprd 227 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F 
~~> r  x  ->  F  ~~> r  y ) )
1312spimev 2102 . . . . . . . . . . 11  |-  ( F  ~~> r  x  ->  E. y  F 
~~> r  y )
1413adantl 468 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  E. y  F 
~~> r  y )
15 rlimdmafv.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : A --> CC )
1615adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  ~~> r  x )  ->  F : A
--> CC )
1716adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F : A --> CC )
18 rlimdmafv.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )
1918adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  ~~> r  x )  ->  sup ( A ,  RR* ,  <  )  = +oo )
2019adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
21 simprl 763 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  y )
22 simprr 765 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  F  ~~> r  z )
2317, 20, 21, 22rlimuni 13607 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  ( F 
~~> r  y  /\  F  ~~> r  z ) )  ->  y  =  z )
2423ex 436 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( ( F 
~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
2524alrimivv 1773 . . . . . . . . . 10  |-  ( (
ph  /\  F  ~~> r  x )  ->  A. y A. z ( ( F  ~~> r  y  /\  F  ~~> r  z )  -> 
y  =  z ) )
26 breq2 4405 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F 
~~> r  y  <->  F  ~~> r  z ) )
2726eu4 2346 . . . . . . . . . 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 669 . . . . . . . . 9  |-  ( (
ph  /\  F  ~~> r  x )  ->  E! y  F 
~~> r  y )
29 dfdfat2 38627 . . . . . . . . 9  |-  (  ~~> r defAt  F  <->  ( F  e.  dom  ~~> r  /\  E! y  F  ~~> r  y ) )
3010, 28, 29sylanbrc 669 . . . . . . . 8  |-  ( (
ph  /\  F  ~~> r  x )  ->  ~~> r defAt  F )
31 afvfundmfveq 38634 . . . . . . . 8  |-  (  ~~> r defAt  F  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F
) )
3230, 31syl 17 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  (  ~~> r  `  F ) )
33 df-fv 5589 . . . . . . . 8  |-  (  ~~> r  `  F )  =  ( iota w F  ~~> r  w )
3415adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F : A --> CC )
3518adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  sup ( A ,  RR* ,  <  )  = +oo )
36 simprr 765 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  w )
37 simprl 763 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  F  ~~> r  x )
3834, 35, 36, 37rlimuni 13607 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( F  ~~> r  x  /\  F  ~~> r  w ) )  ->  w  =  x )
3938expr 619 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  ->  w  =  x ) )
40 breq2 4405 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  ( F 
~~> r  w  <->  F  ~~> r  x ) )
413, 40syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( w  =  x  ->  F  ~~> r  w ) )
4239, 41impbid 194 . . . . . . . . . . 11  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( F  ~~> r  w  <->  w  =  x
) )
4342adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( F 
~~> r  w  <->  w  =  x ) )
4443iota5 5565 . . . . . . . . 9  |-  ( ( ( ph  /\  F  ~~> r  x )  /\  x  e.  _V )  ->  ( iota w F  ~~> r  w )  =  x )
457, 44mpan2 676 . . . . . . . 8  |-  ( (
ph  /\  F  ~~> r  x )  ->  ( iota w F  ~~> r  w )  =  x )
4633, 45syl5eq 2496 . . . . . . 7  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r  `  F )  =  x )
4732, 46eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  F  ~~> r  x )  ->  (  ~~> r ''' F )  =  x )
483, 47breqtrrd 4428 . . . . 5  |-  ( (
ph  /\  F  ~~> r  x )  ->  F  ~~> r  (  ~~> r ''' F ) )
4948ex 436 . . . 4  |-  ( ph  ->  ( F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
5049exlimdv 1778 . . 3  |-  ( ph  ->  ( E. x  F  ~~> r  x  ->  F  ~~> r  (  ~~> r ''' F ) ) )
512, 50syl5 33 . 2  |-  ( ph  ->  ( F  e.  dom  ~~> r  ->  F  ~~> r  (  ~~> r ''' F ) ) )
524releldmi 5070 . 2  |-  ( F  ~~> r  (  ~~> r ''' F )  ->  F  e.  dom  ~~> r  )
5351, 52impbid1 207 1  |-  ( ph  ->  ( F  e.  dom  ~~> r  <-> 
F  ~~> r  (  ~~> r ''' F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1441    = wceq 1443   E.wex 1662    e. wcel 1886   E!weu 2298   _Vcvv 3044   class class class wbr 4401   dom cdm 4833   iotacio 5543   -->wf 5577   ` cfv 5581   supcsup 7951   CCcc 9534   +oocpnf 9669   RR*cxr 9671    < clt 9672    ~~> r crli 13542   defAt wdfat 38608  '''cafv 38609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-rlim 13546  df-dfat 38611  df-afv 38612
This theorem is referenced by: (None)
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