Theorem coass 3512

Description: Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations.

Assertion

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

Proof of Theorem coass

Step	Hyp	Ref	Expression
1		relco 3484	. 2 |- Rel ((A o. B) o. C)
2		relco 3484	. 2 |- Rel (A o. (B o. C))
3		excom 1046	. . . 4 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.wE.z(xCz /\ (zBw /\ wAy)))
4		anass 439	. . . . 5 |- (((xCz /\ zBw) /\ wAy) <-> (xCz /\ (zBw /\ wAy)))
5	4	2exbii 1052	. . . 4 |- (E.wE.z((xCz /\ zBw) /\ wAy) <-> E.wE.z(xCz /\ (zBw /\ wAy)))
6	3, 5	bitr4 176	. . 3 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.wE.z((xCz /\ zBw) /\ wAy))
7		df-br 2620	. . . . . . 7 |- (z(A o. B)y <-> <.z, y>. e. (A o. B))
8		visset 1813	. . . . . . . 8 |- z e. V
9		visset 1813	. . . . . . . 8 |- y e. V
10	8, 9	opelco 3288	. . . . . . 7 |- (<.z, y>. e. (A o. B) <-> E.w(zBw /\ wAy))
11	7, 10	bitr 173	. . . . . 6 |- (z(A o. B)y <-> E.w(zBw /\ wAy))
12	11	anbi2i 480	. . . . 5 |- ((xCz /\ z(A o. B)y) <-> (xCz /\ E.w(zBw /\ wAy)))
13	12	exbii 1051	. . . 4 |- (E.z(xCz /\ z(A o. B)y) <-> E.z(xCz /\ E.w(zBw /\ wAy)))
14		visset 1813	. . . . 5 |- x e. V
15	14, 9	opelco 3288	. . . 4 |- (<.x, y>. e. ((A o. B) o. C) <-> E.z(xCz /\ z(A o. B)y))
16		19.42v 1308	. . . . 5 |- (E.w(xCz /\ (zBw /\ wAy)) <-> (xCz /\ E.w(zBw /\ wAy)))
17	16	exbii 1051	. . . 4 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.z(xCz /\ E.w(zBw /\ wAy)))
18	13, 15, 17	3bitr4 183	. . 3 |- (<.x, y>. e. ((A o. B) o. C) <-> E.zE.w(xCz /\ (zBw /\ wAy)))
19		df-br 2620	. . . . . . 7 |- (x(B o. C)w <-> <.x, w>. e. (B o. C))
20		visset 1813	. . . . . . . 8 |- w e. V
21	14, 20	opelco 3288	. . . . . . 7 |- (<.x, w>. e. (B o. C) <-> E.z(xCz /\ zBw))
22	19, 21	bitr 173	. . . . . 6 |- (x(B o. C)w <-> E.z(xCz /\ zBw))
23	22	anbi1i 481	. . . . 5 |- ((x(B o. C)w /\ wAy) <-> (E.z(xCz /\ zBw) /\ wAy))
24	23	exbii 1051	. . . 4 |- (E.w(x(B o. C)w /\ wAy) <-> E.w(E.z(xCz /\ zBw) /\ wAy))
25	14, 9	opelco 3288	. . . 4 |- (<.x, y>. e. (A o. (B o. C)) <-> E.w(x(B o. C)w /\ wAy))
26		19.41v 1305	. . . . 5 |- (E.z((xCz /\ zBw) /\ wAy) <-> (E.z(xCz /\ zBw) /\ wAy))
27	26	exbii 1051	. . . 4 |- (E.wE.z((xCz /\ zBw) /\ wAy) <-> E.w(E.z(xCz /\ zBw) /\ wAy))
28	24, 25, 27	3bitr4 183	. . 3 |- (<.x, y>. e. (A o. (B o. C)) <-> E.wE.z((xCz /\ zBw) /\ wAy))
29	6, 18, 28	3bitr4 183	. 2 |- (<.x, y>. e. ((A o. B) o. C) <-> <.x, y>. e. (A o. (B o. C)))
30	1, 2, 29	eqrelriv 3251	1 |- ((A o. B) o. C) = (A o. (B o. C))

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411   class class class wbr 2619   o. ccom 3174

This theorem is referenced by:  mapenlem1 4489  mapenlem2 4490  pjsdi2 10085  pjadj2co 10132  pj3lem1 10134  pj3 10136  symggrpi 10406  hmeogrp 10538

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

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