Theorem resieq 4970

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 4049	. . . . 5 |- ( x = C -> ( B ( _I |` A ) x <-> B ( _I |` A ) C ) )
2		eqeq2 2215	. . . . 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 2775	. . . . 5 |- x e. _V
6	5	opres 4969	. . . 4 |- ( B e. A -> ( <. B , x >. e. ( _I |` A ) <-> <. B , x >. e. _I ) )
7		df-br 4046	. . . 4 |- ( B ( _I |` A ) x <-> <. B , x >. e. ( _I |` A ) )
8	5	ideq 4831	. . . . 5 |- ( B _I x <-> B = x )
9		df-br 4046	. . . . 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 2833	. 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 1373   e. wcel 2176   <.cop 3636   class class class wbr 4045   _I cid 4336   |` cres 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-res 4688

This theorem is referenced by:  foeqcnvco  5861  f1eqcocnv  5862