Theorem opelresi 5283

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 5279	. 2 |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> ( <. A , A >. e. _I /\ A e. B ) ) )
2		ididg 5154	. . . 4 |- ( A e. V -> A _I A )
3		df-br 4448	. . . 4 |- ( A _I A <-> <. A , A >. e. _I )
4	2, 3	sylib 196	. . 3 |- ( A e. V -> <. A , A >. e. _I )
5	4	biantrurd 508	. 2 |- ( A e. V -> ( A e. B <-> ( <. A , A >. e. _I /\ A e. B ) ) )
6	1, 5	bitr4d 256	1 |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   <.cop 4033   class class class wbr 4447   _I cid 4790   |` cres 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-res 5011

This theorem is referenced by:  issref  5378  ustfilxp  20450  ustelimasn  20460  metustfbasOLD  20803  metustfbas  20804