Theorem limcfval 23636

Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypotheses

Ref	Expression
limcval.j	|- J = ( K ↾t ( A u. { B } ) )
limcval.k	|- K = ( TopOpen ` ℂfld )

Assertion

Ref	Expression
limcfval	|- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> ( ( F lim CC B ) = { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } /\ ( F lim CC B ) C_ CC ) )

Distinct variable groups:   y, z, A   y, B, z   y, F, z   y, K, z   y, J

Allowed substitution hint:   J( z)

Proof of Theorem limcfval

Dummy variables f j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-limc 23630	. . . 4 |- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y | [. ( TopOpen ` ℂfld ) / j ]. ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) } )
2	1	a1i 11	. . 3 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y | [. ( TopOpen ` ℂfld ) / j ]. ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) } ) )
3		fvexd 6203	. . . . 5 |- ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) -> ( TopOpen ` ℂfld ) e. _V )
4		simplrl 800	. . . . . . . . . 10 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> f = F )
5	4	dmeqd 5326	. . . . . . . . 9 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> dom f = dom F )
6		simpll1 1100	. . . . . . . . . 10 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> F : A --> CC )
7		fdm 6051	. . . . . . . . . 10 |- ( F : A --> CC -> dom F = A )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> dom F = A )
9	5, 8	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> dom f = A )
10		simplrr 801	. . . . . . . . 9 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> x = B )
11	10	sneqd 4189	. . . . . . . 8 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> { x } = { B } )
12	9, 11	uneq12d 3768	. . . . . . 7 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( dom f u. { x } ) = ( A u. { B } ) )
13	10	eqeq2d 2632	. . . . . . . 8 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( z = x <-> z = B ) )
14	4	fveq1d 6193	. . . . . . . 8 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( f ` z ) = ( F ` z ) )
15	13, 14	ifbieq2d 4111	. . . . . . 7 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> if ( z = x , y , ( f ` z ) ) = if ( z = B , y , ( F ` z ) ) )
16	12, 15	mpteq12dv 4733	. . . . . 6 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) = ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) )
17		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> j = ( TopOpen ` ℂfld ) )
18		limcval.k	. . . . . . . . . . 11 |- K = ( TopOpen ` ℂfld )
19	17, 18	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> j = K )
20	19, 12	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( j ↾t ( dom f u. { x } ) ) = ( K ↾t ( A u. { B } ) ) )
21		limcval.j	. . . . . . . . 9 |- J = ( K ↾t ( A u. { B } ) )
22	20, 21	syl6eqr 2674	. . . . . . . 8 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( j ↾t ( dom f u. { x } ) ) = J )
23	22, 19	oveq12d 6668	. . . . . . 7 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( ( j ↾t ( dom f u. { x } ) ) CnP j ) = ( J CnP K ) )
24	23, 10	fveq12d 6197	. . . . . 6 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) = ( ( J CnP K ) ` B ) )
25	16, 24	eleq12d 2695	. . . . 5 |- ( ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) /\ j = ( TopOpen ` ℂfld ) ) -> ( ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) <-> ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) ) )
26	3, 25	sbcied 3472	. . . 4 |- ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) -> ( [. ( TopOpen ` ℂfld ) / j ]. ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) <-> ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) ) )
27	26	abbidv 2741	. . 3 |- ( ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) /\ ( f = F /\ x = B ) ) -> { y | [. ( TopOpen ` ℂfld ) / j ]. ( z e. ( dom f u. { x } ) |-> if ( z = x , y , ( f ` z ) ) ) e. ( ( ( j ↾t ( dom f u. { x } ) ) CnP j ) ` x ) } = { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } )
28		cnex 10017	. . . . 5 |- CC e. _V
29		elpm2r 7875	. . . . 5 |- ( ( ( CC e. _V /\ CC e. _V ) /\ ( F : A --> CC /\ A C_ CC ) ) -> F e. ( CC ^pm CC ) )
30	28, 28, 29	mpanl12 718	. . . 4 |- ( ( F : A --> CC /\ A C_ CC ) -> F e. ( CC ^pm CC ) )
31	30	3adant3 1081	. . 3 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> F e. ( CC ^pm CC ) )
32		simp3 1063	. . 3 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> B e. CC )
33		eqid 2622	. . . . . 6 |- ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) = ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) )
34	21, 18, 33	limcvallem 23635	. . . . 5 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> ( ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) -> y e. CC ) )
35	34	abssdv 3676	. . . 4 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } C_ CC )
36	28	ssex 4802	. . . 4 |- ( { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } C_ CC -> { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } e. _V )
37	35, 36	syl 17	. . 3 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } e. _V )
38	2, 27, 31, 32, 37	ovmpt2d 6788	. 2 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> ( F lim CC B ) = { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } )
39	38, 35	eqsstrd 3639	. 2 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> ( F lim CC B ) C_ CC )
40	38, 39	jca 554	1 |- ( ( F : A --> CC /\ A C_ CC /\ B e. CC ) -> ( ( F lim CC B ) = { y | ( z e. ( A u. { B } ) |-> if ( z = B , y , ( F ` z ) ) ) e. ( ( J CnP K ) ` B ) } /\ ( F lim CC B ) C_ CC ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   u. cun 3572   C_ wss 3574   ifcif 4086   {csn 4177   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^pm cpm 7858   CCcc 9934   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   CnP ccnp 21029   lim CC climc 23626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cnp 21032  df-xms 22125  df-ms 22126  df-limc 23630

This theorem is referenced by:  ellimc  23637  limccl  23639