Theorem tailf 32370

Description: The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Hypothesis

Ref	Expression
tailf.1	|- X = dom D

Assertion

Ref	Expression
tailf	|- ( D e. DirRel -> ( tail ` D ) : X --> ~P X )

Proof of Theorem tailf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5477	. . . . . . 7 |- ( D " { x } ) C_ ran D
2		ssun2 3777	. . . . . . . 8 |- ran D C_ ( dom D u. ran D )
3		dmrnssfld 5384	. . . . . . . 8 |- ( dom D u. ran D ) C_ U. U. D
4	2, 3	sstri 3612	. . . . . . 7 |- ran D C_ U. U. D
5	1, 4	sstri 3612	. . . . . 6 |- ( D " { x } ) C_ U. U. D
6		tailf.1	. . . . . . 7 |- X = dom D
7		dirdm 17234	. . . . . . 7 |- ( D e. DirRel -> dom D = U. U. D )
8	6, 7	syl5req 2669	. . . . . 6 |- ( D e. DirRel -> U. U. D = X )
9	5, 8	syl5sseq 3653	. . . . 5 |- ( D e. DirRel -> ( D " { x } ) C_ X )
10		dmexg 7097	. . . . . . 7 |- ( D e. DirRel -> dom D e. _V )
11	6, 10	syl5eqel 2705	. . . . . 6 |- ( D e. DirRel -> X e. _V )
12		elpw2g 4827	. . . . . 6 |- ( X e. _V -> ( ( D " { x } ) e. ~P X <-> ( D " { x } ) C_ X ) )
13	11, 12	syl 17	. . . . 5 |- ( D e. DirRel -> ( ( D " { x } ) e. ~P X <-> ( D " { x } ) C_ X ) )
14	9, 13	mpbird 247	. . . 4 |- ( D e. DirRel -> ( D " { x } ) e. ~P X )
15	14	ralrimivw 2967	. . 3 |- ( D e. DirRel -> A. x e. X ( D " { x } ) e. ~P X )
16		eqid 2622	. . . 4 |- ( x e. X |-> ( D " { x } ) ) = ( x e. X |-> ( D " { x } ) )
17	16	fmpt 6381	. . 3 |- ( A. x e. X ( D " { x } ) e. ~P X <-> ( x e. X |-> ( D " { x } ) ) : X --> ~P X )
18	15, 17	sylib 208	. 2 |- ( D e. DirRel -> ( x e. X |-> ( D " { x } ) ) : X --> ~P X )
19	6	tailfval 32367	. . 3 |- ( D e. DirRel -> ( tail ` D ) = ( x e. X |-> ( D " { x } ) ) )
20	19	feq1d 6030	. 2 |- ( D e. DirRel -> ( ( tail ` D ) : X --> ~P X <-> ( x e. X |-> ( D " { x } ) ) : X --> ~P X ) )
21	18, 20	mpbird 247	1 |- ( D e. DirRel -> ( tail ` D ) : X --> ~P X )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888   DirRelcdir 17228   tailctail 17229

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-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-dir 17230  df-tail 17231

This theorem is referenced by:  tailfb  32372  filnetlem4  32376