Theorem resieq 4901

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 3993	. . . . 5 |- ( x = C -> ( B ( _I |` A ) x <-> B ( _I |` A ) C ) )
2		eqeq2 2180	. . . . 5 |- ( x = C -> ( B = x <-> B = C ) )
3	1, 2	bibi12d 234	. . . 4 |- ( x = C -> ( ( B ( _I |` A ) x <-> B = x ) <-> ( B ( _I |` A ) C <-> B = C ) ) )
4	3	imbi2d 229	. . 3 |- ( x = C -> ( ( B e. A -> ( B ( _I |` A ) x <-> B = x ) ) <-> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) ) )
5		vex 2733	. . . . 5 |- x e. _V
6	5	opres 4900	. . . 4 |- ( B e. A -> ( <. B , x >. e. ( _I |` A ) <-> <. B , x >. e. _I ) )
7		df-br 3990	. . . 4 |- ( B ( _I |` A ) x <-> <. B , x >. e. ( _I |` A ) )
8	5	ideq 4763	. . . . 5 |- ( B _I x <-> B = x )
9		df-br 3990	. . . . 5 |- ( B _I x <-> <. B , x >. e. _I )
10	8, 9	bitr3i 185	. . . 4 |- ( B = x <-> <. B , x >. e. _I )
11	6, 7, 10	3bitr4g 222	. . 3 |- ( B e. A -> ( B ( _I |` A ) x <-> B = x ) )
12	4, 11	vtoclg 2790	. 2 |- ( C e. A -> ( B e. A -> ( B ( _I |` A ) C <-> B = C ) ) )
13	12	impcom 124	1 |- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989   _I cid 4273   |` cres 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  foeqcnvco  5769  f1eqcocnv  5770