Theorem tailfval 32367

Description: The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Hypothesis

Ref	Expression
tailfval.1	|- X = dom D

Assertion

Ref	Expression
tailfval	|- ( D e. DirRel -> ( tail ` D ) = ( x e. X |-> ( D " { x } ) ) )

Distinct variable groups:   x, D   x, X

Proof of Theorem tailfval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexg 6955	. . . 4 |- ( D e. DirRel -> U. D e. _V )
2		uniexg 6955	. . . 4 |- ( U. D e. _V -> U. U. D e. _V )
3		mptexg 6484	. . . 4 |- ( U. U. D e. _V -> ( x e. U. U. D |-> ( D " { x } ) ) e. _V )
4	1, 2, 3	3syl 18	. . 3 |- ( D e. DirRel -> ( x e. U. U. D |-> ( D " { x } ) ) e. _V )
5		unieq 4444	. . . . . 6 |- ( d = D -> U. d = U. D )
6	5	unieqd 4446	. . . . 5 |- ( d = D -> U. U. d = U. U. D )
7		imaeq1 5461	. . . . 5 |- ( d = D -> ( d " { x } ) = ( D " { x } ) )
8	6, 7	mpteq12dv 4733	. . . 4 |- ( d = D -> ( x e. U. U. d |-> ( d " { x } ) ) = ( x e. U. U. D |-> ( D " { x } ) ) )
9		df-tail 17231	. . . 4 |- tail = ( d e. DirRel |-> ( x e. U. U. d |-> ( d " { x } ) ) )
10	8, 9	fvmptg 6280	. . 3 |- ( ( D e. DirRel /\ ( x e. U. U. D |-> ( D " { x } ) ) e. _V ) -> ( tail ` D ) = ( x e. U. U. D |-> ( D " { x } ) ) )
11	4, 10	mpdan 702	. 2 |- ( D e. DirRel -> ( tail ` D ) = ( x e. U. U. D |-> ( D " { x } ) ) )
12		tailfval.1	. . . 4 |- X = dom D
13		dirdm 17234	. . . 4 |- ( D e. DirRel -> dom D = U. U. D )
14	12, 13	syl5req 2669	. . 3 |- ( D e. DirRel -> U. U. D = X )
15	14	mpteq1d 4738	. 2 |- ( D e. DirRel -> ( x e. U. U. D |-> ( D " { x } ) ) = ( x e. X |-> ( D " { x } ) ) )
16	11, 15	eqtrd 2656	1 |- ( D e. DirRel -> ( tail ` D ) = ( x e. X |-> ( D " { x } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   "cima 5117   ` 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-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:  tailval  32368  tailf  32370