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Theorem mptun 6693
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 5231 . 2 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
2 df-mpt 5231 . . . 4 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3 df-mpt 5231 . . . 4 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
42, 3uneq12i 4160 . . 3 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
5 elun 4147 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 624 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶))
7 andir 1007 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
86, 7bitri 274 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
98opabbii 5214 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
10 unopab 5229 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
119, 10eqtr4i 2763 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
124, 11eqtr4i 2763 . 2 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
131, 12eqtr4i 2763 1 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 845   = wceq 1541  wcel 2106  cun 3945  {copab 5209  cmpt 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-opab 5210  df-mpt 5231
This theorem is referenced by:  partfun  6694  fmptap  7164  fmptapd  7165  esumrnmpt2  33054  ptrest  36475  fsuppssind  41162
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