Theorem eliniseg 5053

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 5051	. . . 4 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> <. B , C >. e. `' A ) )
3		df-br 4046	. . . 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 4860	. . 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 2176   _Vcvv 2772   {csn 3633   <.cop 3636   class class class wbr 4045   `'ccnv 4675   "cima 4679

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by:  epini  5054  iniseg  5055  dfco2a  5184  isoini  5889  pilem3  15288