Theorem opelresi 5051

Description: <. A , A >. belongs to a restriction of the identity class iff A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)

Assertion

Ref	Expression
opelresi	|- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) )

Proof of Theorem opelresi

Step	Hyp	Ref	Expression
1		opelresg 5047	. 2 |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> ( <. A , A >. e. _I /\ A e. B ) ) )
2		ididg 4910	. . . 4 |- ( A e. V -> A _I A )
3		df-br 4112	. . . 4 |- ( A _I A <-> <. A , A >. e. _I )
4	2, 3	sylib 122	. . 3 |- ( A e. V -> <. A , A >. e. _I )
5	4	biantrurd 305	. 2 |- ( A e. V -> ( A e. B <-> ( <. A , A >. e. _I /\ A e. B ) ) )
6	1, 5	bitr4d 191	1 |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   <.cop 3694   class class class wbr 4111   _I cid 4411   |` cres 4753

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 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-res 4763

This theorem is referenced by:  issref  5147