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Theorem dmpropg 4736
 Description: The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmpropg ((B 𝑉 𝐷 𝑊) → dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = {A, 𝐶})

Proof of Theorem dmpropg
StepHypRef Expression
1 dmsnopg 4735 . . 3 (B 𝑉 → dom {⟨A, B⟩} = {A})
2 dmsnopg 4735 . . 3 (𝐷 𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3 uneq12 3086 . . 3 ((dom {⟨A, B⟩} = {A} dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨A, B⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({A} ∪ {𝐶}))
41, 2, 3syl2an 273 . 2 ((B 𝑉 𝐷 𝑊) → (dom {⟨A, B⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({A} ∪ {𝐶}))
5 df-pr 3374 . . . 4 {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩})
65dmeqi 4479 . . 3 dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩})
7 dmun 4485 . . 3 dom ({⟨A, B⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨A, B⟩} ∪ dom {⟨𝐶, 𝐷⟩})
86, 7eqtri 2057 . 2 dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨A, B⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9 df-pr 3374 . 2 {A, 𝐶} = ({A} ∪ {𝐶})
104, 8, 93eqtr4g 2094 1 ((B 𝑉 𝐷 𝑊) → dom {⟨A, B⟩, ⟨𝐶, 𝐷⟩} = {A, 𝐶})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ∪ cun 2909  {csn 3367  {cpr 3368  ⟨cop 3370  dom cdm 4288 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298 This theorem is referenced by:  dmprop  4738  funtpg  4893  fnprg  4897
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