Theorem resco 4402

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 4242	. 2 |- Rel ((A o. B) |` C)
2		relco 4392	. 2 |- Rel (A o. (B |` C))
3		visset 2295	. . . . . 6 |- x e. _V
4		visset 2295	. . . . . 6 |- y e. _V
5	3, 4	brco 4132	. . . . 5 |- (x(A o. B)y <-> E.z(xBz /\ zAy))
6	5	anbi1i 539	. . . 4 |- ((x(A o. B)y /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
7		19.41v 1685	. . . 4 |- (E.z((xBz /\ zAy) /\ x e. C) <-> (E.z(xBz /\ zAy) /\ x e. C))
8		an23 543	. . . . . 6 |- (((xBz /\ zAy) /\ x e. C) <-> ((xBz /\ x e. C) /\ zAy))
9		visset 2295	. . . . . . . 8 |- z e. _V
10	9	brres 4223	. . . . . . 7 |- (x(B |` C)z <-> (xBz /\ x e. C))
11	10	anbi1i 539	. . . . . 6 |- ((x(B |` C)z /\ zAy) <-> ((xBz /\ x e. C) /\ zAy))
12	8, 11	bitr4i 193	. . . . 5 |- (((xBz /\ zAy) /\ x e. C) <-> (x(B |` C)z /\ zAy))
13	12	exbii 1398	. . . 4 |- (E.z((xBz /\ zAy) /\ x e. C) <-> E.z(x(B |` C)z /\ zAy))
14	6, 7, 13	3bitr2i 196	. . 3 |- ((x(A o. B)y /\ x e. C) <-> E.z(x(B |` C)z /\ zAy))
15	4	brres 4223	. . 3 |- (x((A o. B) |` C)y <-> (x(A o. B)y /\ x e. C))
16	3, 4	brco 4132	. . 3 |- (x(A o. (B |` C))y <-> E.z(x(B |` C)z /\ zAy))
17	14, 15, 16	3bitr4i 200	. 2 |- (x((A o. B) |` C)y <-> x(A o. (B |` C))y)
18	1, 2, 17	eqbrriv 4082	1 |- ((A o. B) |` C) = (A o. (B |` C))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   class class class wbr 3338   |` cres 3988   o. ccom 3990

This theorem is referenced by:  cocnvcnv2 4409  hhssims 10778  cmprelid2 14447  cmpdia 14453  cocnv 15716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-co 4003  df-res 4006

Copyright terms: Public domain