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.

Step	Hyp	Ref	Expression
1		eldmg 5133	. . . 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 13073	. . . . . . . . . . . 12 |- Rel ~~> r
5	4	brrelexi 4977	. . . . . . . . . . 11 |- ( F ~~> r x -> F e. _V )
6	5	adantl 466	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> F e. _V )
7		vex 3071	. . . . . . . . . . 11 |- x e. _V
8	7	a1i 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 )
10	6, 8, 3, 9	syl3anc 1219	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> F e. dom ~~> r )
11		breq2 4394	. . . . . . . . . . . . 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 1963	. . . . . . . . . . 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 13130	. . . . . . . . . . . 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 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 ) )
27	26	eu4 2325	. . . . . . . . . 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 30175	. . . . . . . . 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 30182	. . . . . . . 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 5524	. . . . . . . 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 13130	. . . . . . . . . . . . 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 4394	. . . . . . . . . . . . 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 5499	. . . . . . . . 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 2504	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ` F ) = x )
47	32, 46	eqtrd 2492	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = x )
48	3, 47	breqtrrd 4416	. . . . 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 1691	. . 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 5174	. 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 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)