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Theorem coundi 4745
 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 3806 . . 3 ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)}) = {⟨x, y⟩ ∣ (z(xBz zAy) z(x𝐶z zAy))}
2 brun 3780 . . . . . . . 8 (x(B𝐶)z ↔ (xBz x𝐶z))
32anbi1i 434 . . . . . . 7 ((x(B𝐶)z zAy) ↔ ((xBz x𝐶z) zAy))
4 andir 720 . . . . . . 7 (((xBz x𝐶z) zAy) ↔ ((xBz zAy) (x𝐶z zAy)))
53, 4bitri 173 . . . . . 6 ((x(B𝐶)z zAy) ↔ ((xBz zAy) (x𝐶z zAy)))
65exbii 1474 . . . . 5 (z(x(B𝐶)z zAy) ↔ z((xBz zAy) (x𝐶z zAy)))
7 19.43 1497 . . . . 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 3794 . . 3 {⟨x, y⟩ ∣ (z(xBz zAy) z(x𝐶z zAy))} = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
101, 9eqtri 2038 . 2 ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)}) = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
11 df-co 4277 . . 3 (AB) = {⟨x, y⟩ ∣ z(xBz zAy)}
12 df-co 4277 . . 3 (A𝐶) = {⟨x, y⟩ ∣ z(x𝐶z zAy)}
1311, 12uneq12i 3068 . 2 ((AB) ∪ (A𝐶)) = ({⟨x, y⟩ ∣ z(xBz zAy)} ∪ {⟨x, y⟩ ∣ z(x𝐶z zAy)})
14 df-co 4277 . 2 (A ∘ (B𝐶)) = {⟨x, y⟩ ∣ z(x(B𝐶)z zAy)}
1510, 13, 143eqtr4ri 2049 1 (A ∘ (B𝐶)) = ((AB) ∪ (A𝐶))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∨ wo 616   = wceq 1226  ∃wex 1358   ∪ cun 2888   class class class wbr 3734  {copab 3787   ∘ ccom 4272 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-br 3735  df-opab 3789  df-co 4277 This theorem is referenced by:  relcoi1  4772
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