Theorem resco 3506

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 3393	. 2 |- Rel ((A o. B) |` C)
2		relco 3490	. 2 |- Rel (A o. (B |` C))
3		visset 1816	. . . . . 6 |- x e. V
4		visset 1816	. . . . . 6 |- y e. V
5	3, 4	brco 3295	. . . . 5 |- (x(A o. B)y <-> E.z(xBz /\ zAy))
6	5	anbi1i 483	. . . 4 |- ((x(A o. B)y /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
7		19.41v 1307	. . . 4 |- (E.z((xBz /\ zAy) /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
8		an23 487	. . . . . 6 |- (((xBz /\ zAy) /\ x e. C) <-> ((xBz /\ x e. C) /\ zAy))
9		visset 1816	. . . . . . . 8 |- z e. V
10	9	brres 3379	. . . . . . 7 |- (x(B |` C)z <-> (xBz /\ x e. C))
11	10	anbi1i 483	. . . . . 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 1053	. . . 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 3379	. . 3 |- (x((A o. B) |` C)y <-> (x(A o. B)y /\ x e. C))
16	3, 4	brco 3295	. . 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 3258	1 |- ((A o. B) |` C) = (A o. (B |` C))

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   class class class wbr 2624   |` cres 3178   o. ccom 3180

This theorem is referenced by:  cocnvcnv2 3512  hhssims 9140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-co 3193  df-res 3196

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