Theorem resieq 4952

Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

Assertion

Ref	Expression
resieq	|- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) )

Proof of Theorem resieq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4033	. . . . 5 |- ( x = C -> ( B ( _I |` A ) x <-> B ( _I |` A ) C ) )
2		eqeq2 2203	. . . . 5 |- ( x = C -> ( B = x <-> B = C ) )
3	1, 2	bibi12d 235	. . . 4 |- ( x = C -> ( ( B ( _I |` A ) x <-> B = x ) <-> ( B ( _I |` A ) C <-> B = C ) ) )
4	3	imbi2d 230	. . 3 |- ( x = C -> ( ( B e. A -> ( B ( _I |` A ) x <-> B = x ) ) <-> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) ) )
5		vex 2763	. . . . 5 |- x e. _V
6	5	opres 4951	. . . 4 |- ( B e. A -> ( <. B , x >. e. ( _I |` A ) <-> <. B , x >. e. _I ) )
7		df-br 4030	. . . 4 |- ( B ( _I |` A ) x <-> <. B , x >. e. ( _I |` A ) )
8	5	ideq 4814	. . . . 5 |- ( B _I x <-> B = x )
9		df-br 4030	. . . . 5 |- ( B _I x <-> <. B , x >. e. _I )
10	8, 9	bitr3i 186	. . . 4 |- ( B = x <-> <. B , x >. e. _I )
11	6, 7, 10	3bitr4g 223	. . 3 |- ( B e. A -> ( B ( _I |` A ) x <-> B = x ) )
12	4, 11	vtoclg 2820	. 2 |- ( C e. A -> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) )
13	12	impcom 125	1 |- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029   _I cid 4319   |` cres 4661

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-res 4671

This theorem is referenced by:  foeqcnvco  5833  f1eqcocnv  5834