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Theorem dmsnopg 5017
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnopg (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Proof of Theorem dmsnopg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2692 . . . . . 6 𝑥 ∈ V
2 vex 2692 . . . . . 6 𝑦 ∈ V
31, 2opth1 4165 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
43exlimiv 1578 . . . 4 (∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5 opeq1 3712 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6 opeq2 3713 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
76eqeq1d 2149 . . . . . 6 (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
87spcegv 2777 . . . . 5 (𝐵𝑉 → (⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩ → ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
95, 8syl5 32 . . . 4 (𝐵𝑉 → (𝑥 = 𝐴 → ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
104, 9impbid2 142 . . 3 (𝐵𝑉 → (∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ 𝑥 = 𝐴))
111eldm2 4744 . . . 4 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
121, 2opex 4158 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 3547 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1413exbii 1585 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1511, 14bitri 183 . . 3 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16 velsn 3548 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1710, 15, 163bitr4g 222 . 2 (𝐵𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
1817eqrdv 2138 1 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wex 1469  wcel 1481  {csn 3531  cop 3534  dom cdm 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-dm 4556
This theorem is referenced by:  dmpropg  5018  dmsnop  5019  rnsnopg  5024  elxp4  5033  fnsng  5177  funprg  5180  funtpg  5181  fntpg  5186  ennnfonelemhdmp1  11956  ennnfonelemkh  11959  setsvala  12027  setsresg  12034  setscom  12036  setsslid  12046  strle1g  12086
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