Theorem resco 3500

Description: Associative law for the restriction of a composition.

Assertion

Ref	Expression
resco	|- ((A o. B) |` C) = (A o. (B |` C))

Proof of Theorem resco

Step	Hyp	Ref	Expression
1		relres 3387	. 2 |- Rel ((A o. B) |` C)
2		relco 3484	. 2 |- Rel (A o. (B |` C))
3		visset 1813	. . . . . 6 |- x e. V
4		visset 1813	. . . . . 6 |- y e. V
5	3, 4	brco 3289	. . . . 5 |- (x(A o. B)y <-> E.z(xBz /\ zAy))
6	5	anbi1i 481	. . . 4 |- ((x(A o. B)y /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
7		19.41v 1305	. . . 4 |- (E.z((xBz /\ zAy) /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
8		an23 485	. . . . . 6 |- (((xBz /\ zAy) /\ x e. C) <-> ((xBz /\ x e. C) /\ zAy))
9		visset 1813	. . . . . . . 8 |- z e. V
10	9	brres 3373	. . . . . . 7 |- (x(B |` C)z <-> (xBz /\ x e. C))
11	10	anbi1i 481	. . . . . 6 |- ((x(B |` C)z /\ zAy) <-> ((xBz /\ x e. C) /\ zAy))
12	8, 11	bitr4 176	. . . . 5 |- (((xBz /\ zAy) /\ x e. C) <-> (x(B |` C)z /\ zAy))
13	12	exbii 1051	. . . 4 |- (E.z((xBz /\ zAy) /\ x e. C) <-> E.z(x(B |` C)z /\ zAy))
14	6, 7, 13	3bitr2 179	. . 3 |- ((x(A o. B)y /\ x e. C) <-> E.z(x(B |` C)z /\ zAy))
15	4	brres 3373	. . 3 |- (x((A o. B) |` C)y <-> (x(A o. B)y /\ x e. C))
16	3, 4	brco 3289	. . 3 |- (x(A o. (B |` C))y <-> E.z(x(B |` C)z /\ zAy))
17	14, 15, 16	3bitr4 183	. 2 |- (x((A o. B) |` C)y <-> x(A o. (B |` C))y)
18	1, 2, 17	eqbrriv 3252	1 |- ((A o. B) |` C) = (A o. (B |` C))

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   class class class wbr 2619   |` cres 3172   o. ccom 3174

This theorem is referenced by:  cocnvcnv2 3506  hhssims 9145

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-xp 3184  df-rel 3185  df-co 3187  df-res 3190

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