Theorem coass 3498

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 ∘ B) ∘ C) = (A ∘ (B ∘ C))

Proof of Theorem coass

Step	Hyp	Ref	Expression
1		relco 3470	. 2 ⊢ Rel ((A ∘ B) ∘ C)
2		relco 3470	. 2 ⊢ Rel (A ∘ (B ∘ C))
3		excom 1042	. . . 4 ⊢ (∃z∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ ∃w∃z(xCz ⋀ (zBw ⋀ wAy)))
4		anass 439	. . . . 5 ⊢ (((xCz ⋀ zBw) ⋀ wAy) ↔ (xCz ⋀ (zBw ⋀ wAy)))
5	4	2exbii 1048	. . . 4 ⊢ (∃w∃z((xCz ⋀ zBw) ⋀ wAy) ↔ ∃w∃z(xCz ⋀ (zBw ⋀ wAy)))
6	3, 5	bitr4 176	. . 3 ⊢ (∃z∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ ∃w∃z((xCz ⋀ zBw) ⋀ wAy))
7		df-br 2610	. . . . . . 7 ⊢ (z(A ∘ B)y ↔ ⟨z, y⟩ ∈ (A ∘ B))
8		visset 1804	. . . . . . . 8 ⊢ z ∈ V
9		visset 1804	. . . . . . . 8 ⊢ y ∈ V
10	8, 9	opelco 3277	. . . . . . 7 ⊢ (⟨z, y⟩ ∈ (A ∘ B) ↔ ∃w(zBw ⋀ wAy))
11	7, 10	bitr 173	. . . . . 6 ⊢ (z(A ∘ B)y ↔ ∃w(zBw ⋀ wAy))
12	11	anbi2i 479	. . . . 5 ⊢ ((xCz ⋀ z(A ∘ B)y) ↔ (xCz ⋀ ∃w(zBw ⋀ wAy)))
13	12	exbii 1047	. . . 4 ⊢ (∃z(xCz ⋀ z(A ∘ B)y) ↔ ∃z(xCz ⋀ ∃w(zBw ⋀ wAy)))
14		visset 1804	. . . . 5 ⊢ x ∈ V
15	14, 9	opelco 3277	. . . 4 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ∃z(xCz ⋀ z(A ∘ B)y))
16		19.42v 1303	. . . . 5 ⊢ (∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ (xCz ⋀ ∃w(zBw ⋀ wAy)))
17	16	exbii 1047	. . . 4 ⊢ (∃z∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ ∃z(xCz ⋀ ∃w(zBw ⋀ wAy)))
18	13, 15, 17	3bitr4 183	. . 3 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ∃z∃w(xCz ⋀ (zBw ⋀ wAy)))
19		df-br 2610	. . . . . . 7 ⊢ (x(B ∘ C)w ↔ ⟨x, w⟩ ∈ (B ∘ C))
20		visset 1804	. . . . . . . 8 ⊢ w ∈ V
21	14, 20	opelco 3277	. . . . . . 7 ⊢ (⟨x, w⟩ ∈ (B ∘ C) ↔ ∃z(xCz ⋀ zBw))
22	19, 21	bitr 173	. . . . . 6 ⊢ (x(B ∘ C)w ↔ ∃z(xCz ⋀ zBw))
23	22	anbi1i 480	. . . . 5 ⊢ ((x(B ∘ C)w ⋀ wAy) ↔ (∃z(xCz ⋀ zBw) ⋀ wAy))
24	23	exbii 1047	. . . 4 ⊢ (∃w(x(B ∘ C)w ⋀ wAy) ↔ ∃w(∃z(xCz ⋀ zBw) ⋀ wAy))
25	14, 9	opelco 3277	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ (B ∘ C)) ↔ ∃w(x(B ∘ C)w ⋀ wAy))
26		19.41v 1300	. . . . 5 ⊢ (∃z((xCz ⋀ zBw) ⋀ wAy) ↔ (∃z(xCz ⋀ zBw) ⋀ wAy))
27	26	exbii 1047	. . . 4 ⊢ (∃w∃z((xCz ⋀ zBw) ⋀ wAy) ↔ ∃w(∃z(xCz ⋀ zBw) ⋀ wAy))
28	24, 25, 27	3bitr4 183	. . 3 ⊢ (⟨x, y⟩ ∈ (A ∘ (B ∘ C)) ↔ ∃w∃z((xCz ⋀ zBw) ⋀ wAy))
29	6, 18, 28	3bitr4 183	. 2 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ⟨x, y⟩ ∈ (A ∘ (B ∘ C)))
30	1, 2, 29	eqrelriv 3241	1 ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C))

Syntax hints:   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∃wex 977  ⟨cop 2401   class class class wbr 2609   ∘ ccom 3164

This theorem is referenced by:  mapenlem1 4469  mapenlem2 4470  pjsdi2 9996  pjadj2co 10042  pj3lem1 10044  pj3 10046  symggrpiOLD 10311  symggrpi 10313  hmeogrp 10425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-co 3177

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