Theorem resieq 4693

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 3826	. . . . 5 |- ( x = C -> ( B ( _I |` A ) x <-> B ( _I |` A ) C ) )
2		eqeq2 2094	. . . . 5 |- ( x = C -> ( B = x <-> B = C ) )
3	1, 2	bibi12d 233	. . . 4 |- ( x = C -> ( ( B ( _I |` A ) x <-> B = x ) <-> ( B ( _I |` A ) C <-> B = C ) ) )
4	3	imbi2d 228	. . 3 |- ( x = C -> ( ( B e. A -> ( B ( _I |` A ) x <-> B = x ) ) <-> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) ) )
5		vex 2618	. . . . 5 |- x e. _V
6	5	opres 4692	. . . 4 |- ( B e. A -> ( <. B , x >. e. ( _I |` A ) <-> <. B , x >. e. _I ) )
7		df-br 3823	. . . 4 |- ( B ( _I |` A ) x <-> <. B , x >. e. ( _I |` A ) )
8	5	ideq 4558	. . . . 5 |- ( B _I x <-> B = x )
9		df-br 3823	. . . . 5 |- ( B _I x <-> <. B , x >. e. _I )
10	8, 9	bitr3i 184	. . . 4 |- ( B = x <-> <. B , x >. e. _I )
11	6, 7, 10	3bitr4g 221	. . 3 |- ( B e. A -> ( B ( _I |` A ) x <-> B = x ) )
12	4, 11	vtoclg 2672	. 2 |- ( C e. A -> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) )
13	12	impcom 123	1 |- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1287   e. wcel 1436   <.cop 3434   class class class wbr 3822   _I cid 4091   |` cres 4415

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-res 4425

This theorem is referenced by:  foeqcnvco  5532  f1eqcocnv  5533