Theorem dmsnop 3317

Description: The domain of a singleton of an ordered pair is the singleton of the first member.

Assertion

Ref	Expression
dmsnop	⊢ dom {⟨A, B⟩} = {A}

Proof of Theorem dmsnop

Step	Hyp	Ref	Expression
1		visset 1804	. . . . . . . . 9 ⊢ x ∈ V
2		visset 1804	. . . . . . . . 9 ⊢ y ∈ V
3	1, 2	opthg 2778	. . . . . . . 8 ⊢ (B ∈ V → (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ⋀ y = B)))
4		opex 2772	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
5	4	elsnc 2421	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
6	3, 5	syl5bb 530	. . . . . . 7 ⊢ (B ∈ V → (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ⋀ y = B)))
7	6	exbidv 1274	. . . . . 6 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ∃y(x = A ⋀ y = B)))
8		19.42v 1303	. . . . . 6 ⊢ (∃y(x = A ⋀ y = B) ↔ (x = A ⋀ ∃y y = B))
9	7, 8	syl6bb 534	. . . . 5 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ⋀ ∃y y = B)))
10		isset 1805	. . . . . 6 ⊢ (B ∈ V ↔ ∃y y = B)
11		iba 640	. . . . . 6 ⊢ (∃y y = B → (x = A ↔ (x = A ⋀ ∃y y = B)))
12	10, 11	sylbi 199	. . . . 5 ⊢ (B ∈ V → (x = A ↔ (x = A ⋀ ∃y y = B)))
13	9, 12	bitr4d 529	. . . 4 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ x = A))
14	13	abbidv 1569	. . 3 ⊢ (B ∈ V → {x∣∃y⟨x, y⟩ ∈ {⟨A, B⟩}} = {x∣x = A})
15		dfdm3 3291	. . 3 ⊢ dom {⟨A, B⟩} = {x∣∃y⟨x, y⟩ ∈ {⟨A, B⟩}}
16		df-sn 2402	. . 3 ⊢ {A} = {x∣x = A}
17	14, 15, 16	3eqtr4g 1523	. 2 ⊢ (B ∈ V → dom {⟨A, B⟩} = {A})
18		opprc2 2490	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
19		sneq 2407	. . . 4 ⊢ (⟨A, B⟩ = ⟨A, A⟩ → {⟨A, B⟩} = {⟨A, A⟩})
20		dmeq 3300	. . . 4 ⊢ ({⟨A, B⟩} = {⟨A, A⟩} → dom {⟨A, B⟩} = dom {⟨A, A⟩})
21	18, 19, 20	3syl 20	. . 3 ⊢ (¬ B ∈ V → dom {⟨A, B⟩} = dom {⟨A, A⟩})
22	1, 2	opthg 2778	. . . . . . . . . 10 ⊢ (A ∈ V → (⟨x, y⟩ = ⟨A, A⟩ ↔ (x = A ⋀ y = A)))
23	4	elsnc 2421	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ {⟨A, A⟩} ↔ ⟨x, y⟩ = ⟨A, A⟩)
24	22, 23	syl5bb 530	. . . . . . . . 9 ⊢ (A ∈ V → (⟨x, y⟩ ∈ {⟨A, A⟩} ↔ (x = A ⋀ y = A)))
25	24	exbidv 1274	. . . . . . . 8 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ ∃y(x = A ⋀ y = A)))
26		19.42v 1303	. . . . . . . 8 ⊢ (∃y(x = A ⋀ y = A) ↔ (x = A ⋀ ∃y y = A))
27	25, 26	syl6bb 534	. . . . . . 7 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ (x = A ⋀ ∃y y = A)))
28		isset 1805	. . . . . . . 8 ⊢ (A ∈ V ↔ ∃y y = A)
29		iba 640	. . . . . . . 8 ⊢ (∃y y = A → (x = A ↔ (x = A ⋀ ∃y y = A)))
30	28, 29	sylbi 199	. . . . . . 7 ⊢ (A ∈ V → (x = A ↔ (x = A ⋀ ∃y y = A)))
31	27, 30	bitr4d 529	. . . . . 6 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ x = A))
32	31	abbidv 1569	. . . . 5 ⊢ (A ∈ V → {x∣∃y⟨x, y⟩ ∈ {⟨A, A⟩}} = {x∣x = A})
33		dfdm3 3291	. . . . 5 ⊢ dom {⟨A, A⟩} = {x∣∃y⟨x, y⟩ ∈ {⟨A, A⟩}}
34	32, 33, 16	3eqtr4g 1523	. . . 4 ⊢ (A ∈ V → dom {⟨A, A⟩} = {A})
35		dmsnsn0 3314	. . . . 5 ⊢ dom {{∅}} = ∅
36		anidm 432	. . . . . . 7 ⊢ ((¬ A ∈ V ⋀ ¬ A ∈ V) ↔ ¬ A ∈ V)
37		opprc3 2787	. . . . . . 7 ⊢ ((¬ A ∈ V ⋀ ¬ A ∈ V) ↔ ⟨A, A⟩ = {∅})
38	36, 37	bitr3 175	. . . . . 6 ⊢ (¬ A ∈ V ↔ ⟨A, A⟩ = {∅})
39		sneq 2407	. . . . . . 7 ⊢ (⟨A, A⟩ = {∅} → {⟨A, A⟩} = {{∅}})
40	39	dmeqd 3302	. . . . . 6 ⊢ (⟨A, A⟩ = {∅} → dom {⟨A, A⟩} = dom {{∅}})
41	38, 40	sylbi 199	. . . . 5 ⊢ (¬ A ∈ V → dom {⟨A, A⟩} = dom {{∅}})
42		snprc 2433	. . . . . 6 ⊢ (¬ A ∈ V ↔ {A} = ∅)
43	42	biimp 151	. . . . 5 ⊢ (¬ A ∈ V → {A} = ∅)
44	35, 41, 43	3eqtr4a 1524	. . . 4 ⊢ (¬ A ∈ V → dom {⟨A, A⟩} = {A})
45	34, 44	pm2.61i 126	. . 3 ⊢ dom {⟨A, A⟩} = {A}
46	21, 45	syl6eq 1515	. 2 ⊢ (¬ B ∈ V → dom {⟨A, B⟩} = {A})
47	17, 46	pm2.61i 126	1 ⊢ dom {⟨A, B⟩} = {A}

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∃wex 977  {cab 1456  Vcvv 1802  ∅c0 2270  {csn 2399  ⟨cop 2401  dom cdm 3160

This theorem is referenced by:  dmsnsnsn 3318  op1sta 3434  rnsnop 3436  f1osn 3704  tfrlem10 3905  ringsn 8100  1alg 10498  1ded 10515  1cat 10536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-dm 3178

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