Theorem dmsnop 2547

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 1350	. . . . . . . . 9 ⊢ x ∈ V
2		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
3	1, 2	opthg 1899	. . . . . . . 8 ⊢ (B ∈ V → (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B)))
4		opex 1893	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
5	4	elsnc 1826	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
6	3, 5	syl5bb 410	. . . . . . 7 ⊢ (B ∈ V → (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ∧ y = B)))
7	6	biexdv 936	. . . . . 6 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ∃y(x = A ∧ y = B)))
8		19.42v 966	. . . . . 6 ⊢ (∃y(x = A ∧ y = B) ↔ (x = A ∧ ∃y y = B))
9	7, 8	syl6bb 414	. . . . 5 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ (x = A ∧ ∃y y = B)))
10		isset 1351	. . . . . 6 ⊢ (B ∈ V ↔ ∃y y = B)
11		iba 486	. . . . . 6 ⊢ (∃y y = B → (x = A ↔ (x = A ∧ ∃y y = B)))
12	10, 11	sylbi 174	. . . . 5 ⊢ (B ∈ V → (x = A ↔ (x = A ∧ ∃y y = B)))
13	9, 12	bitr4d 409	. . . 4 ⊢ (B ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, B⟩} ↔ x = A))
14	13	biabdv 1183	. . 3 ⊢ (B ∈ V → {x∣∃y⟨x, y⟩ ∈ {⟨A, B⟩}} = {x∣x = A})
15		dfdm3 2522	. . 3 ⊢ dom {⟨A, B⟩} = {x∣∃y⟨x, y⟩ ∈ {⟨A, B⟩}}
16		df-sn 1811	. . 3 ⊢ {A} = {x∣x = A}
17	14 15, 16	3eqtr4g 1147	. 2 ⊢ (B ∈ V → dom {⟨A, B⟩} = {A})
18		opprc2 1907	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
19		sneq 1816	. . . 4 ⊢ (⟨A, B⟩ = ⟨A, A⟩ → {⟨A, B⟩} = {⟨A, A⟩})
20		dmeq 2531	. . . 4 ⊢ ({⟨A, B⟩} = {⟨A, A⟩} → dom {⟨A, B⟩} = dom {⟨A, A⟩})
21	18, 19, 20	3syl 21	. . 3 ⊢ (¬ B ∈ V → dom {⟨A, B⟩} = dom {⟨A, A⟩})
22	1, 2	opthg 1899	. . . . . . . . . 10 ⊢ (A ∈ V → (⟨x, y⟩ = ⟨A, A⟩ ↔ (x = A ∧ y = A)))
23	4	elsnc 1826	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ {⟨A, A⟩} ↔ ⟨x, y⟩ = ⟨A, A⟩)
24	22, 23	syl5bb 410	. . . . . . . . 9 ⊢ (A ∈ V → (⟨x, y⟩ ∈ {⟨A, A⟩} ↔ (x = A ∧ y = A)))
25	24	biexdv 936	. . . . . . . 8 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ ∃y(x = A ∧ y = A)))
26		19.42v 966	. . . . . . . 8 ⊢ (∃y(x = A ∧ y = A) ↔ (x = A ∧ ∃y y = A))
27	25, 26	syl6bb 414	. . . . . . 7 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ (x = A ∧ ∃y y = A)))
28		isset 1351	. . . . . . . 8 ⊢ (A ∈ V ↔ ∃y y = A)
29		iba 486	. . . . . . . 8 ⊢ (∃y y = A → (x = A ↔ (x = A ∧ ∃y y = A)))
30	28, 29	sylbi 174	. . . . . . 7 ⊢ (A ∈ V → (x = A ↔ (x = A ∧ ∃y y = A)))
31	27, 30	bitr4d 409	. . . . . 6 ⊢ (A ∈ V → (∃y⟨x, y⟩ ∈ {⟨A, A⟩} ↔ x = A))
32	31	biabdv 1183	. . . . 5 ⊢ (A ∈ V → {x∣∃y⟨x, y⟩ ∈ {⟨A, A⟩}} = {x∣x = A})
33		dfdm3 2522	. . . . 5 ⊢ dom {⟨A, A⟩} = {x∣∃y⟨x, y⟩ ∈ {⟨A, A⟩}}
34	32, 33, 16	3eqtr4g 1147	. . . 4 ⊢ (A ∈ V → dom {⟨A, A⟩} = {A})
35		anidm 331	. . . . . . . 8 ⊢ ((¬ A ∈ V ∧ ¬ A ∈ V) ↔ ¬ A ∈ V)
36		opprc3 1908	. . . . . . . 8 ⊢ ((¬ A ∈ V ∧ ¬ A ∈ V) ↔ ⟨A, A⟩ = {∅})
37	35, 36	bitr3 153	. . . . . . 7 ⊢ (¬ A ∈ V ↔ ⟨A, A⟩ = {∅})
38		sneq 1816	. . . . . . . 8 ⊢ (⟨A, A⟩ = {∅} → {⟨A, A⟩} = {{∅}})
39	38	dmeqd 2533	. . . . . . 7 ⊢ (⟨A, A⟩ = {∅} → dom {⟨A, A⟩} = dom {{∅}})
40	37, 39	sylbi 174	. . . . . 6 ⊢ (¬ A ∈ V → dom {⟨A, A⟩} = dom {{∅}})
41		dmsnsn0 2544	. . . . . 6 ⊢ dom {{∅}} = ∅
42	40, 41	syl6eq 1140	. . . . 5 ⊢ (¬ A ∈ V → dom {⟨A, A⟩} = ∅)
43		snprc 1838	. . . . . 6 ⊢ (¬ A ∈ V ↔ {A} = ∅)
44	43	biimp 133	. . . . 5 ⊢ (¬ A ∈ V → {A} = ∅)
45	42, 44	eqtr4d 1131	. . . 4 ⊢ (¬ A ∈ V → dom {⟨A, A⟩} = {A})
46	34, 45	pm2.61i 110	. . 3 ⊢ dom {⟨A, A⟩} = {A}
47	21, 46	syl6eq 1140	. 2 ⊢ (¬ B ∈ V → dom {⟨A, B⟩} = {A})
48	17, 47	pm2.61i 110	1 ⊢ dom {⟨A, B⟩} = {A}

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ⟨cop 1810  dom cdm 2410

This theorem is referenced by:  dmsnsnsn 2548  op1sta 2635  rnsnop 2637  f1osn 2827  tfrlem10 2958  fac0 4871

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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