Theorem coass 3504

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 3476	. 2 ⊢ Rel ((A ∘ B) ∘ C)
2		relco 3476	. 2 ⊢ Rel (A ∘ (B ∘ C))
3		excom 1044	. . . 4 ⊢ (∃z∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ ∃w∃z(xCz ⋀ (zBw ⋀ wAy)))
4		anass 439	. . . . 5 ⊢ (((xCz ⋀ zBw) ⋀ wAy) ↔ (xCz ⋀ (zBw ⋀ wAy)))
5	4	2exbii 1050	. . . 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 2615	. . . . . . 7 ⊢ (z(A ∘ B)y ↔ ⟨z, y⟩ ∈ (A ∘ B))
8		visset 1809	. . . . . . . 8 ⊢ z ∈ V
9		visset 1809	. . . . . . . 8 ⊢ y ∈ V
10	8, 9	opelco 3283	. . . . . . 7 ⊢ (⟨z, y⟩ ∈ (A ∘ B) ↔ ∃w(zBw ⋀ wAy))
11	7, 10	bitr 173	. . . . . 6 ⊢ (z(A ∘ B)y ↔ ∃w(zBw ⋀ wAy))
12	11	anbi2i 480	. . . . 5 ⊢ ((xCz ⋀ z(A ∘ B)y) ↔ (xCz ⋀ ∃w(zBw ⋀ wAy)))
13	12	exbii 1049	. . . 4 ⊢ (∃z(xCz ⋀ z(A ∘ B)y) ↔ ∃z(xCz ⋀ ∃w(zBw ⋀ wAy)))
14		visset 1809	. . . . 5 ⊢ x ∈ V
15	14, 9	opelco 3283	. . . 4 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ∃z(xCz ⋀ z(A ∘ B)y))
16		19.42v 1306	. . . . 5 ⊢ (∃w(xCz ⋀ (zBw ⋀ wAy)) ↔ (xCz ⋀ ∃w(zBw ⋀ wAy)))
17	16	exbii 1049	. . . 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 2615	. . . . . . 7 ⊢ (x(B ∘ C)w ↔ ⟨x, w⟩ ∈ (B ∘ C))
20		visset 1809	. . . . . . . 8 ⊢ w ∈ V
21	14, 20	opelco 3283	. . . . . . 7 ⊢ (⟨x, w⟩ ∈ (B ∘ C) ↔ ∃z(xCz ⋀ zBw))
22	19, 21	bitr 173	. . . . . 6 ⊢ (x(B ∘ C)w ↔ ∃z(xCz ⋀ zBw))
23	22	anbi1i 481	. . . . 5 ⊢ ((x(B ∘ C)w ⋀ wAy) ↔ (∃z(xCz ⋀ zBw) ⋀ wAy))
24	23	exbii 1049	. . . 4 ⊢ (∃w(x(B ∘ C)w ⋀ wAy) ↔ ∃w(∃z(xCz ⋀ zBw) ⋀ wAy))
25	14, 9	opelco 3283	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ (B ∘ C)) ↔ ∃w(x(B ∘ C)w ⋀ wAy))
26		19.41v 1303	. . . . 5 ⊢ (∃z((xCz ⋀ zBw) ⋀ wAy) ↔ (∃z(xCz ⋀ zBw) ⋀ wAy))
27	26	exbii 1049	. . . 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 3246	1 ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C))

Syntax hints:   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978  ⟨cop 2407   class class class wbr 2614   ∘ ccom 3169

This theorem is referenced by:  mapenlem1 4475  mapenlem2 4476  pjsdi2 10023  pjadj2co 10070  pj3lem1 10072  pj3 10074  symggrpi 10340  hmeogrp 10461

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-co 3182

Copyright terms: Public domain