Theorem limcrcl 15381

Description: Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)

Assertion

Ref	Expression
limcrcl	|- ( C e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )

Proof of Theorem limcrcl

Dummy variables d e f x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-limced 15379	. . 3 |- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y e. CC | ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) } )
2	1	elmpocl 6216	. 2 |- ( C e. ( F lim CC B ) -> ( F e. ( CC ^pm CC ) /\ B e. CC ) )
3		cnex 8155	. . . . 5 |- CC e. _V
4	3, 3	elpm2 6848	. . . 4 |- ( F e. ( CC ^pm CC ) <-> ( F : dom F --> CC /\ dom F C_ CC ) )
5	4	anbi1i 458	. . 3 |- ( ( F e. ( CC ^pm CC ) /\ B e. CC ) <-> ( ( F : dom F --> CC /\ dom F C_ CC ) /\ B e. CC ) )
6		df-3an 1006	. . 3 |- ( ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) <-> ( ( F : dom F --> CC /\ dom F C_ CC ) /\ B e. CC ) )
7	5, 6	bitr4i 187	. 2 |- ( ( F e. ( CC ^pm CC ) /\ B e. CC ) <-> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
8	2, 7	sylib 122	1 |- ( C e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   e. wcel 2202   A.wral 2510   E.wrex 2511   {crab 2514   C_ wss 3200   class class class wbr 4088   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6017   ^pm cpm 6817   CCcc 8029   < clt 8213   - cmin 8349   # cap 8760   RR+crp 9887   abscabs 11557   lim CC climc 15377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pm 6819  df-limced 15379

This theorem is referenced by:  limccl  15382  limcdifap  15385  limcimolemlt  15387  limcresi  15389  limccnpcntop  15398  limccnp2lem  15399  limccnp2cntop  15400  limccoap  15401