Theorem imasn 2616

Description: Image of a singleton.

Assertion

Ref	Expression
imasn	⊢ (Rel R → (R “ {A}) = {y∣⟨A, y⟩ ∈ R})

Distinct variable group(s):   y,A   y,R

Proof of Theorem imasn

Step	Hyp	Ref	Expression
1		snprc 1838	. . . . . . 7 ⊢ (¬ A ∈ V ↔ {A} = ∅)
2		imaeq2 2603	. . . . . . 7 ⊢ ({A} = ∅ → (R “ {A}) = (R “ ∅))
3	1, 2	sylbi 174	. . . . . 6 ⊢ (¬ A ∈ V → (R “ {A}) = (R “ ∅))
4		ima0 2615	. . . . . 6 ⊢ (R “ ∅) = ∅
5	3, 4	syl6eq 1140	. . . . 5 ⊢ (¬ A ∈ V → (R “ {A}) = ∅)
6	5	adantl 305	. . . 4 ⊢ ((Rel R ∧ ¬ A ∈ V) → (R “ {A}) = ∅)
7		df-rel 2425	. . . . . . . . . 10 ⊢ (Rel R ↔ R ⊆ (V × V))
8		ssel 1502	. . . . . . . . . 10 ⊢ (R ⊆ (V × V) → (⟨A, y⟩ ∈ R → ⟨A, y⟩ ∈ (V × V)))
9	7, 8	sylbi 174	. . . . . . . . 9 ⊢ (Rel R → (⟨A, y⟩ ∈ R → ⟨A, y⟩ ∈ (V × V)))
10		opelxpex 2445	. . . . . . . . 9 ⊢ (⟨A, y⟩ ∈ (V × V) → A ∈ V)
11	9, 10	syl6 23	. . . . . . . 8 ⊢ (Rel R → (⟨A, y⟩ ∈ R → A ∈ V))
12	11	con3d 87	. . . . . . 7 ⊢ (Rel R → (¬ A ∈ V → ¬ ⟨A, y⟩ ∈ R))
13	12	imp 277	. . . . . 6 ⊢ ((Rel R ∧ ¬ A ∈ V) → ¬ ⟨A, y⟩ ∈ R)
14	13	nexdv 983	. . . . 5 ⊢ ((Rel R ∧ ¬ A ∈ V) → ¬ ∃y⟨A, y⟩ ∈ R)
15		abn0 1715	. . . . . 6 ⊢ (¬ {y∣⟨A, y⟩ ∈ R} = ∅ ↔ ∃y⟨A, y⟩ ∈ R)
16	15	bicon1i 193	. . . . 5 ⊢ (¬ ∃y⟨A, y⟩ ∈ R ↔ {y∣⟨A, y⟩ ∈ R} = ∅)
17	14, 16	sylib 173	. . . 4 ⊢ ((Rel R ∧ ¬ A ∈ V) → {y∣⟨A, y⟩ ∈ R} = ∅)
18	6, 17	eqtr4d 1131	. . 3 ⊢ ((Rel R ∧ ¬ A ∈ V) → (R “ {A}) = {y∣⟨A, y⟩ ∈ R})
19	18	exp 291	. 2 ⊢ (Rel R → (¬ A ∈ V → (R “ {A}) = {y∣⟨A, y⟩ ∈ R}))
20		opeq1 1876	. . . . . . 7 ⊢ (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21	20	eleq1d 1155	. . . . . 6 ⊢ (x = A → (⟨x, y⟩ ∈ R ↔ ⟨A, y⟩ ∈ R))
22	21	ceqsexgv 1412	. . . . 5 ⊢ (A ∈ V → (∃x(x = A ∧ ⟨x, y⟩ ∈ R) ↔ ⟨A, y⟩ ∈ R))
23		elsn 1820	. . . . . . 7 ⊢ (x ∈ {A} ↔ x = A)
24	23	anbi1i 368	. . . . . 6 ⊢ ((x ∈ {A} ∧ ⟨x, y⟩ ∈ R) ↔ (x = A ∧ ⟨x, y⟩ ∈ R))
25	24	biex 733	. . . . 5 ⊢ (∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R) ↔ ∃x(x = A ∧ ⟨x, y⟩ ∈ R))
26	22, 25	syl5bb 410	. . . 4 ⊢ (A ∈ V → (∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R) ↔ ⟨A, y⟩ ∈ R))
27	26	biabdv 1183	. . 3 ⊢ (A ∈ V → {y∣∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R)} = {y∣⟨A, y⟩ ∈ R})
28		dfima3 2605	. . 3 ⊢ (R “ {A}) = {y∣∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R)}
29	27, 28	syl5eq 1136	. 2 ⊢ (A ∈ V → (R “ {A}) = {y∣⟨A, y⟩ ∈ R})
30	19, 29	pm2.61d2 111	1 ⊢ (Rel R → (R “ {A}) = {y∣⟨A, y⟩ ∈ R})

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808  ⟨cop 1810   × cxp 2408   “ cima 2413  Rel wrel 2415

This theorem is referenced by:  fnsnfv 2861  funfv2 2863  mapsn 3269  aceq3 3556

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-rex 1206  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-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

metamath.org