Theorem ideqg 5273

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ideqg	|- ( B e. V -> ( A _I B <-> A = B ) )

Proof of Theorem ideqg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( B e. V -> B e. V )
2		reli 5249	. . . 4 |- Rel _I
3	2	brrelexi 5158	. . 3 |- ( A _I B -> A e. _V )
4	1, 3	anim12ci 591	. 2 |- ( ( B e. V /\ A _I B ) -> ( A e. _V /\ B e. V ) )
5		eleq1 2689	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
6	5	biimparc 504	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
7	6	elexd 3214	. . 3 |- ( ( B e. V /\ A = B ) -> A e. _V )
8		simpl 473	. . 3 |- ( ( B e. V /\ A = B ) -> B e. V )
9	7, 8	jca 554	. 2 |- ( ( B e. V /\ A = B ) -> ( A e. _V /\ B e. V ) )
10		eqeq1 2626	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
11		eqeq2 2633	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
12		df-id 5024	. . 3 |- _I = { <. x , y >. | x = y }
13	10, 11, 12	brabg 4994	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
14	4, 9, 13	pm5.21nd 941	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   _I cid 5023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121

This theorem is referenced by:  ideq  5274  ididg  5275  restidsingOLD  5459  poleloe  5527  isof1oidb  6574  pltval  16960  tglngne  25445  tgelrnln  25525  opeldifid  29412  ideq2  34078  idinxpss  34083  inxpssidinxp  34086  idinxpssinxp  34087  fourierdlem42  40366