Theorem eliniseg 5494

Description: Membership in an initial segment. The idiom ( `' A " { B } ), meaning { x | x A B }, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
eliniseg.1	|- C e. _V

Assertion

Ref	Expression
eliniseg	|- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) )

Proof of Theorem eliniseg

Step	Hyp	Ref	Expression
1		eliniseg.1	. 2 |- C e. _V
2		elimasng 5491	. . . 4 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> <. B , C >. e. `' A ) )
3		df-br 4654	. . . 4 |- ( B `' A C <-> <. B , C >. e. `' A )
4	2, 3	syl6bbr 278	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> B `' A C ) )
5		brcnvg 5303	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( B `' A C <-> C A B ) )
6	4, 5	bitrd 268	. 2 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> C A B ) )
7	1, 6	mpan2 707	1 |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   class class class wbr 4653   `'ccnv 5113   "cima 5117

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  epini  5495  iniseg  5496  dfco2a  5635  elpred  5693  isomin  6587  isoini  6588  fnse  7294  infxpenlem  8836  fpwwe2lem8  9459  fpwwe2lem12  9463  fpwwe2lem13  9464  fpwwe2  9465  canth4  9469  canthwelem  9472  pwfseqlem4  9484  fz1isolem  13245  itg1addlem4  23466  elnlfn  28787  pw2f1ocnv  37604