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Theorem dmsnopg 6212
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 3478 . . . . . 6 𝑥 ∈ V
2 vex 3478 . . . . . 6 𝑦 ∈ V
31, 2opth1 5475 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
43exlimiv 1933 . . . 4 (∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5 opeq1 4873 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6 opeq2 4874 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
76eqeq1d 2734 . . . . . 6 (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
87spcegv 3587 . . . . 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 5901 . . . 4 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
12 opex 5464 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 4643 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1413exbii 1850 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1511, 14bitri 274 . . 3 (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16 velsn 4644 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1710, 15, 163bitr4g 313 . 2 (𝐵𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
1817eqrdv 2730 1 (𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wex 1781  wcel 2106  {csn 4628  cop 4634  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-dm 5686
This theorem is referenced by:  dmsnopss  6213  dmpropg  6214  dmsnop  6215  rnsnopg  6220  fnsng  6600  funprg  6602  funtpg  6603  fntpg  6608  funsnfsupp  9386  s1dmALT  14558  setsval  17099  setsdm  17102  estrreslem2  18089  snstriedgval  28295  1loopgrvd0  28758  1hevtxdg0  28759  1hevtxdg1  28760  1egrvtxdg1  28763  p1evtxdeqlem  28766  wlkp1  28935  eupthp1  29466  trlsegvdeglem5  29474  cosnopne  31911  bnj96  33871  bnj535  33896  ovnovollem1  45362
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