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Theorem imacok 4282
 Description: Image under a composition. (Contributed by SF, 4-Feb-2015.)
Assertion
Ref Expression
imacok ((A k B) “k C) = (Ak (Bk C))

Proof of Theorem imacok
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6 x V
2 vex 2862 . . . . . 6 z V
31, 2opkelcok 4262 . . . . 5 (⟪x, z (A k B) ↔ y(⟪x, y B y, z A))
43rexbii 2639 . . . 4 (x Cx, z (A k B) ↔ x C y(⟪x, y B y, z A))
5 rexcom4 2878 . . . 4 (x C y(⟪x, y B y, z A) ↔ yx C (⟪x, y B y, z A))
6 df-rex 2620 . . . . 5 (y (Bk C)⟪y, z Ay(y (Bk C) y, z A))
7 vex 2862 . . . . . . . . 9 y V
87elimak 4259 . . . . . . . 8 (y (Bk C) ↔ x Cx, y B)
98anbi1i 676 . . . . . . 7 ((y (Bk C) y, z A) ↔ (x Cx, y B y, z A))
10 r19.41v 2764 . . . . . . 7 (x C (⟪x, y B y, z A) ↔ (x Cx, y B y, z A))
119, 10bitr4i 243 . . . . . 6 ((y (Bk C) y, z A) ↔ x C (⟪x, y B y, z A))
1211exbii 1582 . . . . 5 (y(y (Bk C) y, z A) ↔ yx C (⟪x, y B y, z A))
136, 12bitr2i 241 . . . 4 (yx C (⟪x, y B y, z A) ↔ y (Bk C)⟪y, z A)
144, 5, 133bitri 262 . . 3 (x Cx, z (A k B) ↔ y (Bk C)⟪y, z A)
152elimak 4259 . . 3 (z ((A k B) “k C) ↔ x Cx, z (A k B))
162elimak 4259 . . 3 (z (Ak (Bk C)) ↔ y (Bk C)⟪y, z A)
1714, 15, 163bitr4i 268 . 2 (z ((A k B) “k C) ↔ z (Ak (Bk C)))
1817eqriv 2350 1 ((A k B) “k C) = (Ak (Bk C))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟪copk 4057   “k cimak 4179   ∘k ccomk 4180 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-sn 4087 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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190 This theorem is referenced by: (None)
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