Theorem dmsnopg 5254

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 2824	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2824	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	opth1 4371	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
4	3	exlimiv 1651	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5		opeq1 3899	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6		opeq2 3900	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
7	6	eqeq1d 2247	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
8	7	spcegv 2913	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩ → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
9	5, 8	syl5 32	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩))
10	4, 9	impbid2 143	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ 𝑥 = 𝐴))
11	1	eldm2 4974	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
12	1, 2	opex 4364	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3721	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
14	13	exbii 1658	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
15	11, 14	bitri 184	. . 3 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16		velsn 3722	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
17	10, 15, 16	3bitr4g 223	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
18	17	eqrdv 2236	1 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Syntax hints:   → wi 4   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {csn 3705  ⟨cop 3708  dom cdm 4769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-dm 4779

This theorem is referenced by:  dmpropg  5255  dmsnop  5256  rnsnopg  5261  elxp4  5270  fnsng  5423  funprg  5426  funtpg  5427  fntpg  5432  s1dmg  11371  ennnfonelemhdmp1  13278  ennnfonelemkh  13281  setsvala  13361  setsresg  13368  setscom  13370  setsslid  13381  bassetsnn  13387  strle1g  13437  umgr1een  16280  1loopgrvd2fi  16460  1loopgrvd0fi  16461  1hevtxdg0fi  16462  1hevtxdg1en  16463  1hegrvtxdg1fi  16464  p1evtxdeqfilem  16466  trlsegvdeglem5  16619