Theorem eliniseg 5098

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 5096	. . . 4 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> <. B , C >. e. `' A ) )
3		df-br 4084	. . . 4 |- ( B `' A C <-> <. B , C >. e. `' A )
4	2, 3	bitr4di 198	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> B `' A C ) )
5		brcnvg 4903	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( B `' A C <-> C A B ) )
6	4, 5	bitrd 188	. 2 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> C A B ) )
7	1, 6	mpan2 425	1 |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   class class class wbr 4083   `'ccnv 4718   "cima 4722

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  epini  5099  iniseg  5100  dfco2a  5229  isoini  5942  pilem3  15457