Theorem coass 2667

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 2658	. 2 ⊢ Rel ((A ∘ B) ∘ C)
2		relco 2658	. 2 ⊢ Rel (A ∘ (B ∘ C))
3		excom 728	. . . 4 ⊢ (∃z∃w(xCz ∧ (zBw ∧ wAy)) ↔ ∃w∃z(xCz ∧ (zBw ∧ wAy)))
4		anass 336	. . . . 5 ⊢ (((xCz ∧ zBw) ∧ wAy) ↔ (xCz ∧ (zBw ∧ wAy)))
5	4	bi2ex 734	. . . 4 ⊢ (∃w∃z((xCz ∧ zBw) ∧ wAy) ↔ ∃w∃z(xCz ∧ (zBw ∧ wAy)))
6	3, 5	bitr4 154	. . 3 ⊢ (∃z∃w(xCz ∧ (zBw ∧ wAy)) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy))
7		df-br 2063	. . . . . . 7 ⊢ (z(A ∘ B)y ↔ ⟨z, y⟩ ∈ (A ∘ B))
8		visset 1350	. . . . . . . 8 ⊢ z ∈ V
9		visset 1350	. . . . . . . 8 ⊢ y ∈ V
10	8, 9	opelco 2509	. . . . . . 7 ⊢ (⟨z, y⟩ ∈ (A ∘ B) ↔ ∃w(zBw ∧ wAy))
11	7, 10	bitr 151	. . . . . 6 ⊢ (z(A ∘ B)y ↔ ∃w(zBw ∧ wAy))
12	11	anbi2i 367	. . . . 5 ⊢ ((xCz ∧ z(A ∘ B)y) ↔ (xCz ∧ ∃w(zBw ∧ wAy)))
13	12	biex 733	. . . 4 ⊢ (∃z(xCz ∧ z(A ∘ B)y) ↔ ∃z(xCz ∧ ∃w(zBw ∧ wAy)))
14		visset 1350	. . . . 5 ⊢ x ∈ V
15	14, 9	opelco 2509	. . . 4 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ∃z(xCz ∧ z(A ∘ B)y))
16		19.42v 966	. . . . 5 ⊢ (∃w(xCz ∧ (zBw ∧ wAy)) ↔ (xCz ∧ ∃w(zBw ∧ wAy)))
17	16	biex 733	. . . 4 ⊢ (∃z∃w(xCz ∧ (zBw ∧ wAy)) ↔ ∃z(xCz ∧ ∃w(zBw ∧ wAy)))
18	13, 15, 17	3bitr4 158	. . 3 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ∃z∃w(xCz ∧ (zBw ∧ wAy)))
19		df-br 2063	. . . . . . 7 ⊢ (x(B ∘ C)w ↔ ⟨x, w⟩ ∈ (B ∘ C))
20		visset 1350	. . . . . . . 8 ⊢ w ∈ V
21	14, 20	opelco 2509	. . . . . . 7 ⊢ (⟨x, w⟩ ∈ (B ∘ C) ↔ ∃z(xCz ∧ zBw))
22	19, 21	bitr 151	. . . . . 6 ⊢ (x(B ∘ C)w ↔ ∃z(xCz ∧ zBw))
23	22	anbi1i 368	. . . . 5 ⊢ ((x(B ∘ C)w ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy))
24	23	biex 733	. . . 4 ⊢ (∃w(x(B ∘ C)w ∧ wAy) ↔ ∃w(∃z(xCz ∧ zBw) ∧ wAy))
25	14, 9	opelco 2509	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ (B ∘ C)) ↔ ∃w(x(B ∘ C)w ∧ wAy))
26		19.41v 963	. . . . 5 ⊢ (∃z((xCz ∧ zBw) ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy))
27	26	biex 733	. . . 4 ⊢ (∃w∃z((xCz ∧ zBw) ∧ wAy) ↔ ∃w(∃z(xCz ∧ zBw) ∧ wAy))
28	24, 25, 27	3bitr4 158	. . 3 ⊢ (⟨x, y⟩ ∈ (A ∘ (B ∘ C)) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy))
29	6, 18, 28	3bitr4 158	. 2 ⊢ (⟨x, y⟩ ∈ ((A ∘ B) ∘ C) ↔ ⟨x, y⟩ ∈ (A ∘ (B ∘ C)))
30	1, 2, 29	cleqreli 2484	1 ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C))

Syntax hints:   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054   ∘ ccom 2414

This theorem is referenced by:  mapenlem1 3384  mapenlem2 3385  pjsdi2 5627  pjadj2co 5656  pj3lem1 5658  pj3 5660

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-co 2427

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