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.

Step	Hyp	Ref	Expression
1		eldmg 5196	. . . 4 |- ( F e. dom ~~> r -> ( F e. dom ~~> r <-> E. x F ~~> r x ) )
2	1	ibi 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
5	4	brrelexi 5039	. . . . . . . . . . 11 |- ( F ~~> r x -> F e. _V )
6	5	adantl 466	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> F e. _V )
7		vex 3116	. . . . . . . . . . 11 |- x e. _V
8	7	a1i 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 )
10	6, 8, 3, 9	syl3anc 1228	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> F e. dom ~~> r )
11		breq2 4451	. . . . . . . . . . . . 13 |- ( y = x -> ( F ~~> r y <-> F ~~> r x ) )
12	11	biimprd 223	. . . . . . . . . . . 12 |- ( y = x -> ( F ~~> r x -> F ~~> r y ) )
13	12	spimev 1979	. . . . . . . . . . 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 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 )
23	17, 20, 21, 22	rlimuni 13332	. . . . . . . . . . . 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 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 ) )
27	26	eu4 2340	. . . . . . . . . 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 31683	. . . . . . . . 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 31690	. . . . . . . 8 |- ( ~~> r defAt F -> ( ~~> r ''' F ) = ( ~~> r ` F ) )
32	30, 31	syl 16	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = ( ~~> r ` F ) )
33		df-fv 5594	. . . . . . . 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 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 )
38	34, 35, 36, 37	rlimuni 13332	. . . . . . . . . . . . 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 4451	. . . . . . . . . . . . 13 |- ( w = x -> ( F ~~> r w <-> F ~~> r x ) )
41	3, 40	syl5ibrcom 222	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( w = x -> F ~~> r w ) )
42	39, 41	impbid 191	. . . . . . . . . . 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 5569	. . . . . . . . 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 2520	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ` F ) = x )
47	32, 46	eqtrd 2508	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = x )
48	3, 47	breqtrrd 4473	. . . . 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 1700	. . 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 5237	. 2 |- ( F ~~> r ( ~~> r ''' F ) -> F e. dom ~~> r )
53	51, 52	impbid1 203	1 |- ( ph -> ( F e. dom ~~> r <-> F ~~> r ( ~~> r ''' F ) ) )

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)