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Theorem coundi 4765
Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundi (A ∘ (B𝐶)) = ((AB) ∪ (A𝐶))

Proof of Theorem coundi
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 3827 . . 3 ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)}) = {⟨x, y⟩ ∣ (z(xBz zAy) z(x𝐶z zAy))}
2 brun 3801 . . . . . . . 8 (x(B𝐶)z ↔ (xBz x𝐶z))
32anbi1i 431 . . . . . . 7 ((x(B𝐶)z zAy) ↔ ((xBz x𝐶z) zAy))
4 andir 731 . . . . . . 7 (((xBz x𝐶z) zAy) ↔ ((xBz zAy) (x𝐶z zAy)))
53, 4bitri 173 . . . . . 6 ((x(B𝐶)z zAy) ↔ ((xBz zAy) (x𝐶z zAy)))
65exbii 1493 . . . . 5 (z(x(B𝐶)z zAy) ↔ z((xBz zAy) (x𝐶z zAy)))
7 19.43 1516 . . . . 5 (z((xBz zAy) (x𝐶z zAy)) ↔ (z(xBz zAy) z(x𝐶z zAy)))
86, 7bitr2i 174 . . . 4 ((z(xBz zAy) z(x𝐶z zAy)) ↔ z(x(B𝐶)z zAy))
98opabbii 3815 . . 3 {⟨x, y⟩ ∣ (z(xBz zAy) z(x𝐶z zAy))} = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
101, 9eqtri 2057 . 2 ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)}) = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
11 df-co 4297 . . 3 (AB) = {⟨x, y⟩ ∣ z(xBz zAy)}
12 df-co 4297 . . 3 (A𝐶) = {⟨x, y⟩ ∣ z(x𝐶z zAy)}
1311, 12uneq12i 3089 . 2 ((AB) ∪ (A𝐶)) = ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)})
14 df-co 4297 . 2 (A ∘ (B𝐶)) = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
1510, 13, 143eqtr4ri 2068 1 (A ∘ (B𝐶)) = ((AB) ∪ (A𝐶))
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:  relcoi1  4792
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