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Theorem mptun 5097
Description: Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
mptun (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))

Proof of Theorem mptun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3867 . 2 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
2 df-mpt 3867 . . . 4 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3 df-mpt 3867 . . . 4 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
42, 3uneq12i 3136 . . 3 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
5 elun 3125 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 446 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶))
7 andir 766 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
86, 7bitri 182 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
98opabbii 3871 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
10 unopab 3883 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
119, 10eqtr4i 2106 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
124, 11eqtr4i 2106 . 2 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
131, 12eqtr4i 2106 1 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  cun 2982  {copab 3864  cmpt 3865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-opab 3866  df-mpt 3867
This theorem is referenced by:  fmptap  5429  fmptapd  5430
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