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Theorem mptun 6165
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 4862 . 2 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
2 df-mpt 4862 . . . 4 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3 df-mpt 4862 . . . 4 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
42, 3uneq12i 3914 . . 3 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
5 elun 3902 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 602 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶))
7 andir 968 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
86, 7bitri 264 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶) ↔ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶)))
98opabbii 4849 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
10 unopab 4860 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦 = 𝐶) ∨ (𝑥𝐵𝑦 = 𝐶))}
119, 10eqtr4i 2795 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
124, 11eqtr4i 2795 . 2 ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦 = 𝐶)}
131, 12eqtr4i 2795 1 (𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 382  wo 826   = wceq 1630  wcel 2144  cun 3719  {copab 4844  cmpt 4861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-opab 4845  df-mpt 4862
This theorem is referenced by:  fmptap  6579  fmptapd  6580  partfun  29809  esumrnmpt2  30464  ptrest  33734
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