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.

Step	Hyp	Ref	Expression
1		eldmg 5029	. . . 4 |- ( F e. dom ~~> r -> ( F e. dom ~~> r <-> E. x F ~~> r x ) )
2	1	ibi 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
5	4	brrelexi 4874	. . . . . . . . . . 11 |- ( F ~~> r x -> F e. _V )
6	5	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> F e. _V )
7		vex 3047	. . . . . . . . . . 11 |- x e. _V
8	7	a1i 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 )
10	6, 8, 3, 9	syl3anc 1267	. . . . . . . . 9 |- ( ( ph /\ F ~~> r x ) -> F e. dom ~~> r )
11		breq2 4405	. . . . . . . . . . . . 13 |- ( y = x -> ( F ~~> r y <-> F ~~> r x ) )
12	11	biimprd 227	. . . . . . . . . . . 12 |- ( y = x -> ( F ~~> r x -> F ~~> r y ) )
13	12	spimev 2102	. . . . . . . . . . 11 |- ( F ~~> r x -> E. y F ~~> r y )
14	13	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ F ~~> r x ) -> E. y F ~~> r y )
15		rlimdmafv.1	. . . . . . . . . . . . . . 15 |- ( ph -> F : A --> CC )
16	15	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> F : A --> CC )
17	16	adantr 467	. . . . . . . . . . . . 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 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ F ~~> r x ) -> sup ( A , RR* , < ) = +oo )
20	19	adantr 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 )
23	17, 20, 21, 22	rlimuni 13607	. . . . . . . . . . . 12 |- ( ( ( ph /\ F ~~> r x ) /\ ( F ~~> r y /\ F ~~> r z ) ) -> y = z )
24	23	ex 436	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( ( F ~~> r y /\ F ~~> r z ) -> y = z ) )
25	24	alrimivv 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 ) )
27	26	eu4 2346	. . . . . . . . . 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 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 ) )
30	10, 28, 29	sylanbrc 669	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ~~> r defAt F )
31		afvfundmfveq 38634	. . . . . . . 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 5589	. . . . . . . 8 |- ( ~~> r ` F ) = ( iota w F ~~> r w )
34	15	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> F : A --> CC )
35	18	adantr 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 )
38	34, 35, 36, 37	rlimuni 13607	. . . . . . . . . . . . 13 |- ( ( ph /\ ( F ~~> r x /\ F ~~> r w ) ) -> w = x )
39	38	expr 619	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w -> w = x ) )
40		breq2 4405	. . . . . . . . . . . . 13 |- ( w = x -> ( F ~~> r w <-> F ~~> r x ) )
41	3, 40	syl5ibrcom 226	. . . . . . . . . . . 12 |- ( ( ph /\ F ~~> r x ) -> ( w = x -> F ~~> r w ) )
42	39, 41	impbid 194	. . . . . . . . . . 11 |- ( ( ph /\ F ~~> r x ) -> ( F ~~> r w <-> w = x ) )
43	42	adantr 467	. . . . . . . . . 10 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( F ~~> r w <-> w = x ) )
44	43	iota5 5565	. . . . . . . . 9 |- ( ( ( ph /\ F ~~> r x ) /\ x e. _V ) -> ( iota w F ~~> r w ) = x )
45	7, 44	mpan2 676	. . . . . . . 8 |- ( ( ph /\ F ~~> r x ) -> ( iota w F ~~> r w ) = x )
46	33, 45	syl5eq 2496	. . . . . . 7 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ` F ) = x )
47	32, 46	eqtrd 2484	. . . . . 6 |- ( ( ph /\ F ~~> r x ) -> ( ~~> r ''' F ) = x )
48	3, 47	breqtrrd 4428	. . . . 5 |- ( ( ph /\ F ~~> r x ) -> F ~~> r ( ~~> r ''' F ) )
49	48	ex 436	. . . 4 |- ( ph -> ( F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
50	49	exlimdv 1778	. . 3 |- ( ph -> ( E. x F ~~> r x -> F ~~> r ( ~~> r ''' F ) ) )
51	2, 50	syl5 33	. 2 |- ( ph -> ( F e. dom ~~> r -> F ~~> r ( ~~> r ''' F ) ) )
52	4	releldmi 5070	. 2 |- ( F ~~> r ( ~~> r ''' F ) -> F e. dom ~~> r )
53	51, 52	impbid1 207	1 |- ( ph -> ( F e. dom ~~> r <-> F ~~> r ( ~~> r ''' F ) ) )

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)