Theorem dmsnopg 4735

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 2554	. . . . . 6 ⊢ x ∈ V
2		vex 2554	. . . . . 6 ⊢ y ∈ V
3	1, 2	opth1 3964	. . . . 5 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → x = A)
4	3	exlimiv 1486	. . . 4 ⊢ (∃y⟨x, y⟩ = ⟨A, B⟩ → x = A)
5		opeq1 3540	. . . . 5 ⊢ (x = A → ⟨x, B⟩ = ⟨A, B⟩)
6		opeq2 3541	. . . . . . 7 ⊢ (y = B → ⟨x, y⟩ = ⟨x, B⟩)
7	6	eqeq1d 2045	. . . . . 6 ⊢ (y = B → (⟨x, y⟩ = ⟨A, B⟩ ↔ ⟨x, B⟩ = ⟨A, B⟩))
8	7	spcegv 2635	. . . . 5 ⊢ (B ∈ 𝑉 → (⟨x, B⟩ = ⟨A, B⟩ → ∃y⟨x, y⟩ = ⟨A, B⟩))
9	5, 8	syl5 28	. . . 4 ⊢ (B ∈ 𝑉 → (x = A → ∃y⟨x, y⟩ = ⟨A, B⟩))
10	4, 9	impbid2 131	. . 3 ⊢ (B ∈ 𝑉 → (∃y⟨x, y⟩ = ⟨A, B⟩ ↔ x = A))
11	1	eldm2 4476	. . . 4 ⊢ (x ∈ dom {⟨A, B⟩} ↔ ∃y⟨x, y⟩ ∈ {⟨A, B⟩})
12	1, 2	opex 3957	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
13	12	elsnc 3390	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
14	13	exbii 1493	. . . 4 ⊢ (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ∃y⟨x, y⟩ = ⟨A, B⟩)
15	11, 14	bitri 173	. . 3 ⊢ (x ∈ dom {⟨A, B⟩} ↔ ∃y⟨x, y⟩ = ⟨A, B⟩)
16		elsn 3382	. . 3 ⊢ (x ∈ {A} ↔ x = A)
17	10, 15, 16	3bitr4g 212	. 2 ⊢ (B ∈ 𝑉 → (x ∈ dom {⟨A, B⟩} ↔ x ∈ {A}))
18	17	eqrdv 2035	1 ⊢ (B ∈ 𝑉 → dom {⟨A, B⟩} = {A})

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {csn 3367  ⟨cop 3370  dom cdm 4288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298

This theorem is referenced by:  dmpropg  4736  dmsnop  4737  rnsnopg  4742  elxp4  4751  fnsng  4890  funprg  4892  funtpg  4893  fntpg  4898