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Theorem coundir 5009
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 3975 . . 3 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))}
2 brun 3947 . . . . . . . 8 (𝑦(𝐴𝐵)𝑧 ↔ (𝑦𝐴𝑧𝑦𝐵𝑧))
32anbi2i 450 . . . . . . 7 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)))
4 andi 790 . . . . . . 7 ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
53, 4bitri 183 . . . . . 6 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
65exbii 1567 . . . . 5 (∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
7 19.43 1590 . . . . 5 (∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
86, 7bitr2i 184 . . . 4 ((∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧))
98opabbii 3963 . . 3 {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
101, 9eqtri 2136 . 2 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
11 df-co 4516 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
12 df-co 4516 . . 3 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
1311, 12uneq12i 3196 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
14 df-co 4516 . 2 ((𝐴𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
1510, 13, 143eqtr4ri 2147 1 ((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 103  wo 680   = wceq 1314  wex 1451  cun 3037   class class class wbr 3897  {copab 3956  ccom 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-br 3898  df-opab 3958  df-co 4516
This theorem is referenced by: (None)
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