Theorem eltail 32369

Description: An element of a tail. (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
eltail	|- ( ( D e. DirRel /\ A e. X /\ B e. C ) -> ( B e. ( ( tail ` D ) ` A ) <-> A D B ) )

Proof of Theorem eltail

Step	Hyp	Ref	Expression
1		tailfval.1	. . . . 5 |- X = dom D
2	1	tailval 32368	. . . 4 |- ( ( D e. DirRel /\ A e. X ) -> ( ( tail ` D ) ` A ) = ( D " { A } ) )
3	2	eleq2d 2687	. . 3 |- ( ( D e. DirRel /\ A e. X ) -> ( B e. ( ( tail ` D ) ` A ) <-> B e. ( D " { A } ) ) )
4	3	3adant3 1081	. 2 |- ( ( D e. DirRel /\ A e. X /\ B e. C ) -> ( B e. ( ( tail ` D ) ` A ) <-> B e. ( D " { A } ) ) )
5		elimasng 5491	. . . 4 |- ( ( A e. X /\ B e. C ) -> ( B e. ( D " { A } ) <-> <. A , B >. e. D ) )
6		df-br 4654	. . . 4 |- ( A D B <-> <. A , B >. e. D )
7	5, 6	syl6bbr 278	. . 3 |- ( ( A e. X /\ B e. C ) -> ( B e. ( D " { A } ) <-> A D B ) )
8	7	3adant1 1079	. 2 |- ( ( D e. DirRel /\ A e. X /\ B e. C ) -> ( B e. ( D " { A } ) <-> A D B ) )
9	4, 8	bitrd 268	1 |- ( ( D e. DirRel /\ A e. X /\ B e. C ) -> ( B e. ( ( tail ` D ) ` A ) <-> A D B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   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:  tailini  32371  tailfb  32372  filnetlem4  32376