Theorem eliniseg 4981

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 4979	. . . 4 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> <. B , C >. e. `' A ) )
3		df-br 3990	. . . 4 |- ( B `' A C <-> <. B , C >. e. `' A )
4	2, 3	bitr4di 197	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> B `' A C ) )
5		brcnvg 4792	. . 3 |- ( ( B e. V /\ C e. _V ) -> ( B `' A C <-> C A B ) )
6	4, 5	bitrd 187	. 2 |- ( ( B e. V /\ C e. _V ) -> ( C e. ( `' A " { B } ) <-> C A B ) )
7	1, 6	mpan2 423	1 |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   {csn 3583   <.cop 3586   class class class wbr 3989   `'ccnv 4610   "cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  epini  4982  iniseg  4983  dfco2a  5111  isoini  5797  pilem3  13498