Theorem opelresi 5220

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 5216	. 2 |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> ( <. A , A >. e. _I /\ A e. B ) ) )
2		ididg 5091	. . . 4 |- ( A e. V -> A _I A )
3		df-br 4391	. . . 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 1758   <.cop 3981   class class class wbr 4390   _I cid 4729   |` cres 4940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-res 4950

This theorem is referenced by:  issref  5309  ustfilxp  19903  ustelimasn  19913  metustfbasOLD  20256  metustfbas  20257