Theorem rlimdmafv 27908

Description: Two ways to express that a function has a limit, analogous to rlimdm 12300. (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 5024	. . . 4 |- ( F e. dom ~~> r -> ( F e. dom ~~> r <-> E. x F ~~> r x ) )
2	1	ibi 233	. . 3 |- ( F e. dom ~~> r -> E. x F ~~> r x )
3		simpr 448	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> F ~~> r x )
4		rlimrel 12242	. . . . . . . . . . . 12 |- Rel ~~> r
5	4	brrelexi 4877	. . . . . . . . . . 11 |- ( F ~~> r x -> F e. _V )
6	5	adantl 453	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> F e. _V )
7		vex 2919	. . . . . . . . . . 11 |- x e. _V
8	7	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> x e. _V )
9		breldmg 5034	. . . . . . . . . 10 |- ( ( F e. _V /\ x e. _V /\ F ~~> r x ) -> F e. dom ~~> r )
10	6, 8, 3, 9	syl3anc 1184	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> F e. dom ~~> r )
11		breq2 4176	. . . . . . . . . . . . 13 |- ( y = x -> ( F ~~> r y <-> F ~~> r x ) )
12	11	biimprd 215	. . . . . . . . . . . 12 |- ( y = x -> ( F ~~> r x -> F ~~> r y ) )
13	12	spimev 1962	. . . . . . . . . . 11 |- ( F ~~> r x -> E. y F ~~> r y )
14	13	adantl 453	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> E. y F ~~> r y )
15		rlimdmafv.1	. . . . . . . . . . . . . . 15 |- ( ph -> F : A --> CC )
16	15	adantr 452	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> F : A --> CC )
17	16	adantr 452	. . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> sup ( A , RR* , < ) = +oo )
20	19	adantr 452	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> sup ( A , RR* , < ) = +oo )
21		simprl 733	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> F ~~> r y )
22		simprr 734	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> F ~~> r z )
23	17, 20, 21, 22	rlimuni 12299	. . . . . . . . . . . 12 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> y = z )
24	23	ex 424	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( ( F ~~> r y /\ F ~~> r z ) -> y = z ) )
25	24	alrimivv 1639	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> A. y A. z ( ( F ~~> r y /\ F ~~> r z ) -> y = z ) )
26		breq2 4176	. . . . . . . . . . 11 |- ( y = z -> ( F ~~> r y <-> F ~~> r z ) )
27	26	eu4 2293	. . . . . . . . . 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 646	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> E! y F ~~> r y )
29		dfdfat2 27862	. . . . . . . . 9 |- ( ~~> r defAt F <-> ( F e. dom ~~> r /\ E! y F ~~> r y ) )
30	10, 28, 29	sylanbrc 646	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ~~> r defAt F )
31		afvfundmfveq 27869	. . . . . . . 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 5421	. . . . . . . 8 |- ( ~~> r ` F ) = ( iota w F ~~> r w )
34	15	adantr 452	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> F : A --> CC )
35	18	adantr 452	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> sup ( A , RR* , < ) = +oo )
36		simprr 734	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> F ~~> r w )
37		simprl 733	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> F ~~> r x )
38	34, 35, 36, 37	rlimuni 12299	. . . . . . . . . . . . 13 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> w = x )
39	38	expr 599	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w -> w = x ) )
40		breq2 4176	. . . . . . . . . . . . 13 |- ( w = x -> ( F ~~> r w <-> F ~~> r x ) )
41	3, 40	syl5ibrcom 214	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( w = x -> F ~~> r w ) )
42	39, 41	impbid 184	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w <-> w = x ) )
43	42	adantr 452	. . . . . . . . . 10 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( F ~~> r w <-> w = x ) )
44	43	iota5 5397	. . . . . . . . 9 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( iota w F ~~> r w ) = x )
45	7, 44	mpan2 653	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ( iota w F ~~> r w ) = x )
46	33, 45	syl5eq 2448	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ` F ) = x )
47	32, 46	eqtrd 2436	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = x )
48	3, 47	breqtrrd 4198	. . . . 5 |- ( ( ph /\ F ~~> r x ) -> F ~~> r ( ~~> r ''' F ) )
49	48	ex 424	. . . 4 |- ( ph -> ( F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
50	49	exlimdv 1643	. . 3 |- ( ph -> ( E. x F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
51	2, 50	syl5 30	. 2 |- ( ph -> ( F e. dom ~~> r -> F ~~> r ( ~~> r ''' F ) ) )
52	4	releldmi 5065	. 2 |- ( F ~~> r ( ~~> r ''' F ) -> F e. dom ~~> r )
53	51, 52	impbid1 195	1 |- ( ph -> ( F e. dom ~~> r <-> F ~~> r ( ~~> r ''' F ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   A.wal 1546   E.wex 1547   = wceq 1649   e. wcel 1721   E!weu 2254   _Vcvv 2916   class class class wbr 4172   dom cdm 4837   iotacio 5375   -->wf 5409   ` cfv 5413   supcsup 7403   CCcc 8944   +oocpnf 9073   RR*cxr 9075   < clt 9076   ~~> r crli 12234   defAt wdfat 27838  '''cafv 27839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-rlim 12238  df-dfat 27841  df-afv 27842