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Theorem dmpropg 5825
 Description: The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmpropg ((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Proof of Theorem dmpropg
StepHypRef Expression
1 dmsnopg 5823 . . 3 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 dmsnopg 5823 . . 3 (𝐷𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3 uneq12 3960 . . 3 ((dom {⟨𝐴, 𝐵⟩} = {𝐴} ∧ dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
41, 2, 3syl2an 590 . 2 ((𝐵𝑉𝐷𝑊) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
5 df-pr 4371 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
65dmeqi 5528 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7 dmun 5534 . . 3 dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
86, 7eqtri 2821 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9 df-pr 4371 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
104, 8, 93eqtr4g 2858 1 ((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ∪ cun 3767  {csn 4368  {cpr 4370  ⟨cop 4374  dom cdm 5312 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-dm 5322 This theorem is referenced by:  dmprop  5827  funtpg  6155  fnprg  6159  hashdmpropge2  13514  s2dmALT  13993  s4dom  14004  estrreslem2  17092  structiedg0val  26257
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