Theorem coass 4226

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 4203	. 2 |- Rel ((A o. B) o. C)
2		relco 4203	. 2 |- Rel (A o. (B o. C))
3		excom 1231	. . . 4 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.wE.z(xCz /\ (zBw /\ wAy)))
4		anass 485	. . . . 5 |- (((xCz /\ zBw) /\ wAy) <-> (xCz /\ (zBw /\ wAy)))
5	4	2exbii 1237	. . . 4 |- (E.wE.z((xCz /\ zBw) /\ wAy) <-> E.wE.z(xCz /\ (zBw /\ wAy)))
6	3, 5	bitr4i 192	. . 3 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.wE.z((xCz /\ zBw) /\ wAy))
7		df-br 3159	. . . . . . 7 |- (z(A o. B)y <-> <.z, y>. e. (A o. B))
8		visset 2128	. . . . . . . 8 |- z e. _V
9		visset 2128	. . . . . . . 8 |- y e. _V
10	8, 9	opelco 3941	. . . . . . 7 |- (<.z, y>. e. (A o. B) <-> E.w(zBw /\ wAy))
11	7, 10	bitri 189	. . . . . 6 |- (z(A o. B)y <-> E.w(zBw /\ wAy))
12	11	anbi2i 535	. . . . 5 |- ((xCz /\ z(A o. B)y) <-> (xCz /\ E.w(zBw /\ wAy)))
13	12	exbii 1236	. . . 4 |- (E.z(xCz /\ z(A o. B)y) <-> E.z(xCz /\ E.w(zBw /\ wAy)))
14		visset 2128	. . . . 5 |- x e. _V
15	14, 9	opelco 3941	. . . 4 |- (<.x, y>. e. ((A o. B) o. C) <-> E.z(xCz /\ z(A o. B)y))
16		19.42v 1526	. . . . 5 |- (E.w(xCz /\ (zBw /\ wAy)) <-> (xCz /\ E.w(zBw /\ wAy)))
17	16	exbii 1236	. . . 4 |- (E.zE.w(xCz /\ (zBw /\ wAy)) <-> E.z(xCz /\ E.w(zBw /\ wAy)))
18	13, 15, 17	3bitr4i 199	. . 3 |- (<.x, y>. e. ((A o. B) o. C) <-> E.zE.w(xCz /\ (zBw /\ wAy)))
19		df-br 3159	. . . . . . 7 |- (x(B o. C)w <-> <.x, w>. e. (B o. C))
20		visset 2128	. . . . . . . 8 |- w e. _V
21	14, 20	opelco 3941	. . . . . . 7 |- (<.x, w>. e. (B o. C) <-> E.z(xCz /\ zBw))
22	19, 21	bitri 189	. . . . . 6 |- (x(B o. C)w <-> E.z(xCz /\ zBw))
23	22	anbi1i 536	. . . . 5 |- ((x(B o. C)w /\ wAy) <-> (E.z(xCz /\ zBw) /\ wAy))
24	23	exbii 1236	. . . 4 |- (E.w(x(B o. C)w /\ wAy) <-> E.w(E.z(xCz /\ zBw) /\ wAy))
25	14, 9	opelco 3941	. . . 4 |- (<.x, y>. e. (A o. (B o. C)) <-> E.w(x(B o. C)w /\ wAy))
26		19.41v 1523	. . . . 5 |- (E.z((xCz /\ zBw) /\ wAy) <-> (E.z(xCz /\ zBw) /\ wAy))
27	26	exbii 1236	. . . 4 |- (E.wE.z((xCz /\ zBw) /\ wAy) <-> E.w(E.z(xCz /\ zBw) /\ wAy))
28	24, 25, 27	3bitr4i 199	. . 3 |- (<.x, y>. e. (A o. (B o. C)) <-> E.wE.z((xCz /\ zBw) /\ wAy))
29	6, 18, 28	3bitr4i 199	. 2 |- (<.x, y>. e. ((A o. B) o. C) <-> <.x, y>. e. (A o. (B o. C)))
30	1, 2, 29	eqrelriv 3891	1 |- ((A o. B) o. C) = (A o. (B o. C))

Syntax hints:   /\ wa 239   = wceq 1136   e. wcel 1138  E.wex 1164  <.cop 2870   class class class wbr 3158   o. ccom 3801

This theorem is referenced by:  mapenlem1 5393  mapenlem2 5394  symggrpi 9998  pjsdi2i 11521  pjadj2coi 11569  pj3lem1 11571  pj3i 11573  hmeogrp 14612  cocnv 15398

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-v 2127  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-nul 2702  df-pw 2859  df-sn 2873  df-pr 2874  df-op 2877  df-br 3159  df-opab 3214  df-xp 3811  df-rel 3812  df-co 3814

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