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Theorem dmpropg 4821
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 4820 . . 3 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 dmsnopg 4820 . . 3 (𝐷𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3 uneq12 3120 . . 3 ((dom {⟨𝐴, 𝐵⟩} = {𝐴} ∧ dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
41, 2, 3syl2an 277 . 2 ((𝐵𝑉𝐷𝑊) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
5 df-pr 3410 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
65dmeqi 4564 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7 dmun 4570 . . 3 dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
86, 7eqtri 2076 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9 df-pr 3410 . 2 {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
104, 8, 93eqtr4g 2113 1 ((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  cun 2943  {csn 3403  {cpr 3404  cop 3406  dom cdm 4373
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  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 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  dmprop  4823  funtpg  4978  fnprg  4982
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