Theorem resieq 4956

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 4037	. . . . 5 |- ( x = C -> ( B ( _I |` A ) x <-> B ( _I |` A ) C ) )
2		eqeq2 2206	. . . . 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 2766	. . . . 5 |- x e. _V
6	5	opres 4955	. . . 4 |- ( B e. A -> ( <. B , x >. e. ( _I |` A ) <-> <. B , x >. e. _I ) )
7		df-br 4034	. . . 4 |- ( B ( _I |` A ) x <-> <. B , x >. e. ( _I |` A ) )
8	5	ideq 4818	. . . . 5 |- ( B _I x <-> B = x )
9		df-br 4034	. . . . 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 2824	. 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 2167   <.cop 3625   class class class wbr 4033   _I cid 4323   |` cres 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-res 4675

This theorem is referenced by:  foeqcnvco  5837  f1eqcocnv  5838