Theorem resieq 3376

Description: A restricted identity relation is equivalent to equality in its domain.

Assertion

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

Proof of Theorem resieq

Step	Hyp	Ref	Expression
1		breq2 2623	. . . . 5 |- (x = C -> (B(I |` A)x <-> B(I |` A)C))
2		eqeq2 1484	. . . . 5 |- (x = C -> (B = x <-> B = C))
3	1, 2	bibi12d 629	. . . 4 |- (x = C -> ((B(I |` A)x <-> B = x) <-> (B(I |` A)C <-> B = C)))
4	3	imbi2d 612	. . 3 |- (x = C -> ((B e. A -> (B(I |` A)x <-> B = x)) <-> (B e. A -> (B(I |` A)C <-> B = C))))
5		visset 1813	. . . . 5 |- x e. V
6	5	opres 3375	. . . 4 |- (B e. A -> (<.B, x>. e. (I |` A) <-> <.B, x>. e. I))
7		df-br 2620	. . . 4 |- (B(I |` A)x <-> <.B, x>. e. (I |` A))
8	5	ideq 3277	. . . . 5 |- (BIx <-> B = x)
9		df-br 2620	. . . . 5 |- (BIx <-> <.B, x>. e. I)
10	8, 9	bitr3 175	. . . 4 |- (B = x <-> <.B, x>. e. I)
11	6, 7, 10	3bitr4g 555	. . 3 |- (B e. A -> (B(I |` A)x <-> B = x))
12	4, 11	vtoclg 1847	. 2 |- (C e. A -> (B e. A -> (B(I |` A)C <-> B = C)))
13	12	impcom 351	1 |- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  <.cop 2411   class class class wbr 2619  Icid 2831   |` cres 3172

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-res 3190

Copyright terms: Public domain