Theorem imacok 4283

Description: Image under a composition. (Contributed by SF, 4-Feb-2015.)

Assertion

Ref	Expression
imacok	⊢ ((A ∘k B) “k C) = (A “k (B “k C))

Proof of Theorem imacok

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . 6 ⊢ x ∈ V
2		vex 2863	. . . . . 6 ⊢ z ∈ V
3	1, 2	opkelcok 4263	. . . . 5 ⊢ (⟪x, z⟫ ∈ (A ∘k B) ↔ ∃y(⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
4	3	rexbii 2640	. . . 4 ⊢ (∃x ∈ C ⟪x, z⟫ ∈ (A ∘k B) ↔ ∃x ∈ C ∃y(⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
5		rexcom4 2879	. . . 4 ⊢ (∃x ∈ C ∃y(⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A) ↔ ∃y∃x ∈ C (⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
6		df-rex 2621	. . . . 5 ⊢ (∃y ∈ (B “k C)⟪y, z⟫ ∈ A ↔ ∃y(y ∈ (B “k C) ∧ ⟪y, z⟫ ∈ A))
7		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
8	7	elimak 4260	. . . . . . . 8 ⊢ (y ∈ (B “k C) ↔ ∃x ∈ C ⟪x, y⟫ ∈ B)
9	8	anbi1i 676	. . . . . . 7 ⊢ ((y ∈ (B “k C) ∧ ⟪y, z⟫ ∈ A) ↔ (∃x ∈ C ⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
10		r19.41v 2765	. . . . . . 7 ⊢ (∃x ∈ C (⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A) ↔ (∃x ∈ C ⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
11	9, 10	bitr4i 243	. . . . . 6 ⊢ ((y ∈ (B “k C) ∧ ⟪y, z⟫ ∈ A) ↔ ∃x ∈ C (⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
12	11	exbii 1582	. . . . 5 ⊢ (∃y(y ∈ (B “k C) ∧ ⟪y, z⟫ ∈ A) ↔ ∃y∃x ∈ C (⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A))
13	6, 12	bitr2i 241	. . . 4 ⊢ (∃y∃x ∈ C (⟪x, y⟫ ∈ B ∧ ⟪y, z⟫ ∈ A) ↔ ∃y ∈ (B “k C)⟪y, z⟫ ∈ A)
14	4, 5, 13	3bitri 262	. . 3 ⊢ (∃x ∈ C ⟪x, z⟫ ∈ (A ∘k B) ↔ ∃y ∈ (B “k C)⟪y, z⟫ ∈ A)
15	2	elimak 4260	. . 3 ⊢ (z ∈ ((A ∘k B) “k C) ↔ ∃x ∈ C ⟪x, z⟫ ∈ (A ∘k B))
16	2	elimak 4260	. . 3 ⊢ (z ∈ (A “k (B “k C)) ↔ ∃y ∈ (B “k C)⟪y, z⟫ ∈ A)
17	14, 15, 16	3bitr4i 268	. 2 ⊢ (z ∈ ((A ∘k B) “k C) ↔ z ∈ (A “k (B “k C)))
18	17	eqriv 2350	1 ⊢ ((A ∘k B) “k C) = (A “k (B “k C))

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟪copk 4058   “_k cimak 4180   ∘_k ccomk 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191

This theorem is referenced by: (None)