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Theorem dmsnopg 6169
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 3451 . . . . . 6 𝑥 ∈ V
2 vex 3451 . . . . . 6 𝑦 ∈ V
31, 2opth1 5436 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
43exlimiv 1934 . . . 4 (∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5 opeq1 4834 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6 opeq2 4835 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
76eqeq1d 2735 . . . . . 6 (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
87spcegv 3558 . . . . 5 (𝐵𝑉 → (⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩ → ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
95, 8syl5 34 . . . 4 (𝐵𝑉 → (𝑥 = 𝐴 → ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
104, 9impbid2 225 . . 3 (𝐵𝑉 → (∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ 𝑥 = 𝐴))
111eldm2 5861 . . . 4 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
12 opex 5425 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 4605 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1413exbii 1851 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1511, 14bitri 275 . . 3 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16 velsn 4606 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1710, 15, 163bitr4g 314 . 2 (𝐵𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
1817eqrdv 2731 1 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1782  wcel 2107  {csn 4590  cop 4596  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-dm 5647
This theorem is referenced by:  dmsnopss  6170  dmpropg  6171  dmsnop  6172  rnsnopg  6177  fnsng  6557  funprg  6559  funtpg  6560  fntpg  6565  funsnfsupp  9337  s1dmALT  14506  setsval  17047  setsdm  17050  estrreslem2  18034  snstriedgval  28038  1loopgrvd0  28501  1hevtxdg0  28502  1hevtxdg1  28503  1egrvtxdg1  28506  p1evtxdeqlem  28509  wlkp1  28678  eupthp1  29209  trlsegvdeglem5  29217  cosnopne  31662  bnj96  33541  bnj535  33566  ovnovollem1  44987
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