Theorem dmsnopg 5208

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 2805	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2805	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	opth1 4328	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
4	3	exlimiv 1646	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴)
5		opeq1 3862	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
6		opeq2 3863	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩)
7	6	eqeq1d 2240	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
8	7	spcegv 2894	. . . . 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 4929	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
12	1, 2	opex 4321	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3685	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
14	13	exbii 1653	. . . 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 3686	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
17	10, 15, 16	3bitr4g 223	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴}))
18	17	eqrdv 2229	1 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {csn 3669  ⟨cop 3672  dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-dm 4735

This theorem is referenced by:  dmpropg  5209  dmsnop  5210  rnsnopg  5215  elxp4  5224  fnsng  5377  funprg  5380  funtpg  5381  fntpg  5386  s1dmg  11201  ennnfonelemhdmp1  13029  ennnfonelemkh  13032  setsvala  13112  setsresg  13119  setscom  13121  setsslid  13132  bassetsnn  13138  strle1g  13188  umgr1een  15975  1loopgrvd2fi  16155  1loopgrvd0fi  16156  1hevtxdg0fi  16157  1hevtxdg1en  16158  1hegrvtxdg1fi  16159  p1evtxdeqfilem  16161  trlsegvdeglem5  16314