Theorem dmsnopg 4719

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 2538	. . . . . 6 ⊢ x ∈ V
2		vex 2538	. . . . . 6 ⊢ y ∈ V
3	1, 2	opth1 3947	. . . . 5 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → x = A)
4	3	exlimiv 1471	. . . 4 ⊢ (∃y⟨x, y⟩ = ⟨A, B⟩ → x = A)
5		opeq1 3523	. . . . 5 ⊢ (x = A → ⟨x, B⟩ = ⟨A, B⟩)
6		opeq2 3524	. . . . . . 7 ⊢ (y = B → ⟨x, y⟩ = ⟨x, B⟩)
7	6	eqeq1d 2030	. . . . . 6 ⊢ (y = B → (⟨x, y⟩ = ⟨A, B⟩ ↔ ⟨x, B⟩ = ⟨A, B⟩))
8	7	spcegv 2618	. . . . 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 4460	. . . 4 ⊢ (x ∈ dom {⟨A, B⟩} ↔ ∃y⟨x, y⟩ ∈ {⟨A, B⟩})
12	1, 2	opex 3940	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
13	12	elsnc 3373	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
14	13	exbii 1478	. . . 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 3365	. . 3 ⊢ (x ∈ {A} ↔ x = A)
17	10, 15, 16	3bitr4g 212	. 2 ⊢ (B ∈ 𝑉 → (x ∈ dom {⟨A, B⟩} ↔ x ∈ {A}))
18	17	eqrdv 2020	1 ⊢ (B ∈ 𝑉 → dom {⟨A, B⟩} = {A})

Syntax hints:   → wi 4   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {csn 3350  ⟨cop 3353  dom cdm 4272

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-dm 4282

This theorem is referenced by:  dmpropg  4720  dmsnop  4721  rnsnopg  4726  elxp4  4735  fnsng  4873  funprg  4875  funtpg  4876  fntpg  4881