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Theorem dmpropg 6047
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 6045 . . 3 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 dmsnopg 6045 . . 3 (𝐷𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3 uneq12 4048 . . 3 ((dom {⟨𝐴, 𝐵⟩} = {𝐴} ∧ dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
41, 2, 3syl2an 599 . 2 ((𝐵𝑉𝐷𝑊) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
5 df-pr 4519 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
65dmeqi 5747 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7 dmun 5753 . . 3 dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
86, 7eqtri 2761 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9 df-pr 4519 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
104, 8, 93eqtr4g 2798 1 ((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  cun 3841  {csn 4516  {cpr 4518  cop 4522  dom cdm 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-dm 5535
This theorem is referenced by:  dmprop  6049  funtpg  6394  fnprg  6398  hashdmpropge2  13935  s2dmALT  14359  s4dom  14370  estrreslem2  17504  structiedg0val  26967
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