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Theorem mptun 5056
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 3847 . 2 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
2 df-mpt 3847 . . . 4 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3 df-mpt 3847 . . . 4 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
42, 3uneq12i 3122 . . 3 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
5 elun 3111 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 439 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶))
7 andir 743 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
86, 7bitri 177 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
98opabbii 3851 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
10 unopab 3863 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
119, 10eqtr4i 2079 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
124, 11eqtr4i 2079 . 2 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
131, 12eqtr4i 2079 1 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 101  wo 639   = wceq 1259  wcel 1409  cun 2942  {copab 3844  cmpt 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-opab 3846  df-mpt 3847
This theorem is referenced by:  fmptap  5380  fmptapd  5381
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