Theorem dmsnopg 6231

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	⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Proof of Theorem dmsnopg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	opth1 5449	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
4	3	exlimiv 1965	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5		opeq1 4843	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6		opeq2 4844	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
7	6	eqeq1d 2765	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
8	7	spcegv 3555	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩ → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
9	5, 8	syl5 35	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
10	4, 9	impbid2 229	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ 𝑥 = 𝐴))
11	1	eldm2 5893	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
12		opex 5437	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 4599	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
14	13	exbii 1881	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
15	11, 14	bitri 278	. . 3 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16		velsn 4600	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
17	10, 15, 16	3bitr4g 317	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
18	17	eqrdv 2761	1 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {csn 4588  ⟨cop 4838  dom cdm 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-dm 5666

This theorem is used by:  dmsnopss  6232  dmpropg  6233  dmsnop  6234  rnsnopg  6239  fnsng  6620  funprg  6622  funtpg  6623  fntpg  6628  funsnfsupp  9419  s1dmALT  14797  setsval  17386  setsdm  17389  estrreslem2  18353  snstriedgval  29662  1loopgrvd0  30131  1hevtxdg0  30132  1hevtxdg1  30133  1egrvtxdg1  30136  p1evtxdeqlem  30139  wlkp1  30306  eupthp1  30863  trlsegvdeglem5  30871  cosnopne  33374  bnj96  35599  bnj535  35624  ovnovollem1  47933