Theorem eliniseg 5132

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 5130	. . . 4 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> <. B , C >. e. `' A ) )
3		df-br 4110	. . . 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 4936	. . 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 2203   _Vcvv 2813   {csn 3689   <.cop 3692   class class class wbr 4109   `'ccnv 4748   "cima 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  epini  5133  iniseg  5134  dfco2a  5263  isoini  5991  pilem3  15648