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.

Step	Hyp	Ref	Expression
1		eldmg 5021	. . . 4 |- ( F e. dom ~~> r -> ( F e. dom ~~> r <-> E. x F ~~> r x ) )
2	1	ibi 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
5	4	brrelexi 4866	. . . . . . . . . . 11 |- ( F ~~> r x -> F e. _V )
6	5	adantl 466	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> F e. _V )
7		vex 3064	. . . . . . . . . . 11 |- x e. _V
8	7	a1i 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 )
10	6, 8, 3, 9	syl3anc 1232	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> F e. dom ~~> r )
11		breq2 4401	. . . . . . . . . . . . 13 |- ( y = x -> ( F ~~> r y <-> F ~~> r x ) )
12	11	biimprd 225	. . . . . . . . . . . 12 |- ( y = x -> ( F ~~> r x -> F ~~> r y ) )
13	12	spimev 2039	. . . . . . . . . . 11 |- ( F ~~> r x -> E. y F ~~> r y )
14	13	adantl 466	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> E. y F ~~> r y )
15		rlimdmafv.1	. . . . . . . . . . . . . . 15 |- ( ph -> F : A --> CC )
16	15	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> F : A --> CC )
17	16	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> F : A --> CC )
18		rlimdmafv.2	. . . . . . . . . . . . . . 15 |- ( ph -> sup ( A , RR* , < ) = +oo )
19	18	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> sup ( A , RR* , < ) = +oo )
20	19	adantr 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 )
23	17, 20, 21, 22	rlimuni 13524	. . . . . . . . . . . 12 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> y = z )
24	23	ex 434	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( ( F ~~> r y /\ F ~~> r z ) -> y = z ) )
25	24	alrimivv 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 ) )
27	26	eu4 2292	. . . . . . . . . 10 |- ( E! y F ~~> r y <-> ( E. y F ~~> r y /\ A. y A. z ( ( F ~~> r y /\ F ~~> r z ) -> y = z ) ) )
28	14, 25, 27	sylanbrc 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 ) )
30	10, 28, 29	sylanbrc 664	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ~~> r defAt F )
31		afvfundmfveq 37604	. . . . . . . 8 |- ( ~~> r defAt F -> ( ~~> r ''' F ) = ( ~~> r ` F ) )
32	30, 31	syl 17	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = ( ~~> r ` F ) )
33		df-fv 5579	. . . . . . . 8 |- ( ~~> r ` F ) = ( iota w F ~~> r w )
34	15	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> F : A --> CC )
35	18	adantr 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 )
38	34, 35, 36, 37	rlimuni 13524	. . . . . . . . . . . . 13 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> w = x )
39	38	expr 615	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w -> w = x ) )
40		breq2 4401	. . . . . . . . . . . . 13 |- ( w = x -> ( F ~~> r w <-> F ~~> r x ) )
41	3, 40	syl5ibrcom 224	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( w = x -> F ~~> r w ) )
42	39, 41	impbid 192	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w <-> w = x ) )
43	42	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( F ~~> r w <-> w = x ) )
44	43	iota5 5555	. . . . . . . . 9 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( iota w F ~~> r w ) = x )
45	7, 44	mpan2 671	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ( iota w F ~~> r w ) = x )
46	33, 45	syl5eq 2457	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ` F ) = x )
47	32, 46	eqtrd 2445	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = x )
48	3, 47	breqtrrd 4423	. . . . 5 |- ( ( ph /\ F ~~> r x ) -> F ~~> r ( ~~> r ''' F ) )
49	48	ex 434	. . . 4 |- ( ph -> ( F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
50	49	exlimdv 1747	. . 3 |- ( ph -> ( E. x F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
51	2, 50	syl5 32	. 2 |- ( ph -> ( F e. dom ~~> r -> F ~~> r ( ~~> r ''' F ) ) )
52	4	releldmi 5062	. 2 |- ( F ~~> r ( ~~> r ''' F ) -> F e. dom ~~> r )
53	51, 52	impbid1 205	1 |- ( ph -> ( F e. dom ~~> r <-> F ~~> r ( ~~> r ''' F ) ) )

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)