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Theorem coundir 4766
 Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundir ((AB) ∘ 𝐶) = ((A𝐶) ∪ (B𝐶))

Proof of Theorem coundir
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 3827 . . 3 ({⟨x, z⟩ ∣ y(x𝐶y yAz)} ∪ {⟨x, z⟩ ∣ y(x𝐶y yBz)}) = {⟨x, z⟩ ∣ (y(x𝐶y yAz) y(x𝐶y yBz))}
2 brun 3801 . . . . . . . 8 (y(AB)z ↔ (yAz yBz))
32anbi2i 430 . . . . . . 7 ((x𝐶y y(AB)z) ↔ (x𝐶y (yAz yBz)))
4 andi 730 . . . . . . 7 ((x𝐶y (yAz yBz)) ↔ ((x𝐶y yAz) (x𝐶y yBz)))
53, 4bitri 173 . . . . . 6 ((x𝐶y y(AB)z) ↔ ((x𝐶y yAz) (x𝐶y yBz)))
65exbii 1493 . . . . 5 (y(x𝐶y y(AB)z) ↔ y((x𝐶y yAz) (x𝐶y yBz)))
7 19.43 1516 . . . . 5 (y((x𝐶y yAz) (x𝐶y yBz)) ↔ (y(x𝐶y yAz) y(x𝐶y yBz)))
86, 7bitr2i 174 . . . 4 ((y(x𝐶y yAz) y(x𝐶y yBz)) ↔ y(x𝐶y y(AB)z))
98opabbii 3815 . . 3 {⟨x, z⟩ ∣ (y(x𝐶y yAz) y(x𝐶y yBz))} = {⟨x, z⟩ ∣ y(x𝐶y y(AB)z)}
101, 9eqtri 2057 . 2 ({⟨x, z⟩ ∣ y(x𝐶y yAz)} ∪ {⟨x, z⟩ ∣ y(x𝐶y yBz)}) = {⟨x, z⟩ ∣ y(x𝐶y y(AB)z)}
11 df-co 4297 . . 3 (A𝐶) = {⟨x, z⟩ ∣ y(x𝐶y yAz)}
12 df-co 4297 . . 3 (B𝐶) = {⟨x, z⟩ ∣ y(x𝐶y yBz)}
1311, 12uneq12i 3089 . 2 ((A𝐶) ∪ (B𝐶)) = ({⟨x, z⟩ ∣ y(x𝐶y yAz)} ∪ {⟨x, z⟩ ∣ y(x𝐶y yBz)})
14 df-co 4297 . 2 ((AB) ∘ 𝐶) = {⟨x, z⟩ ∣ y(x𝐶y y(AB)z)}
1510, 13, 143eqtr4ri 2068 1 ((AB) ∘ 𝐶) = ((A𝐶) ∪ (B𝐶))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∨ wo 628   = wceq 1242  ∃wex 1378   ∪ cun 2909   class class class wbr 3755  {copab 3808   ∘ ccom 4292 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-br 3756  df-opab 3810  df-co 4297 This theorem is referenced by: (None)
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