Theorem fsn 2895

Description: A function maps a singleton to a singleton iff it is the singleton of a ordered pair.

Hypotheses

Ref	Expression
fsn.1	⊢ A ∈ V
fsn.2	⊢ B ∈ V

Assertion

Ref	Expression
fsn	⊢ (F:{A}–→{B} ↔ F = {⟨A, B⟩})

Proof of Theorem fsn

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
2	1	opelf 2762	. . . . . . . 8 ⊢ ((F:{A}–→{B} ∧ ⟨x, y⟩ ∈ F) → (x ∈ {A} ∧ y ∈ {B}))
3		elsn 1820	. . . . . . . . 9 ⊢ (x ∈ {A} ↔ x = A)
4		elsn 1820	. . . . . . . . 9 ⊢ (y ∈ {B} ↔ y = B)
5	3, 4	anbi12i 369	. . . . . . . 8 ⊢ ((x ∈ {A} ∧ y ∈ {B}) ↔ (x = A ∧ y = B))
6	2, 5	sylib 173	. . . . . . 7 ⊢ ((F:{A}–→{B} ∧ ⟨x, y⟩ ∈ F) → (x = A ∧ y = B))
7	6	exp 291	. . . . . 6 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F → (x = A ∧ y = B)))
8		opeq12 1878	. . . . . . . . 9 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
9	8	eleq1d 1155	. . . . . . . 8 ⊢ ((x = A ∧ y = B) → (⟨x, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
10		fsn.1	. . . . . . . . . . 11 ⊢ A ∈ V
11	10	snid 1830	. . . . . . . . . 10 ⊢ A ∈ {A}
12		feu 2767	. . . . . . . . . 10 ⊢ ((F:{A}–→{B} ∧ A ∈ {A}) → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
13	11, 12	mpan2 519	. . . . . . . . 9 ⊢ (F:{A}–→{B} → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
14		fsn.2	. . . . . . . . . . . 12 ⊢ B ∈ V
15	14	eueq1 1428	. . . . . . . . . . 11 ⊢ ∃!y y = B
16	15	biantru 543	. . . . . . . . . 10 ⊢ (⟨A, B⟩ ∈ F ↔ (⟨A, B⟩ ∈ F ∧ ∃!y y = B))
17		euanv 1053	. . . . . . . . . . 11 ⊢ (∃!y(⟨A, B⟩ ∈ F ∧ y = B) ↔ (⟨A, B⟩ ∈ F ∧ ∃!y y = B))
18		opeq2 1877	. . . . . . . . . . . . . . . 16 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
19	18	eleq1d 1155	. . . . . . . . . . . . . . 15 ⊢ (y = B → (⟨A, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
20	19	pm5.32i 489	. . . . . . . . . . . . . 14 ⊢ ((y = B ∧ ⟨A, y⟩ ∈ F) ↔ (y = B ∧ ⟨A, B⟩ ∈ F))
21	4	anbi1i 368	. . . . . . . . . . . . . 14 ⊢ ((y ∈ {B} ∧ ⟨A, y⟩ ∈ F) ↔ (y = B ∧ ⟨A, y⟩ ∈ F))
22		ancom 333	. . . . . . . . . . . . . 14 ⊢ ((⟨A, B⟩ ∈ F ∧ y = B) ↔ (y = B ∧ ⟨A, B⟩ ∈ F))
23	20, 21, 22	3bitr4r 159	. . . . . . . . . . . . 13 ⊢ ((⟨A, B⟩ ∈ F ∧ y = B) ↔ (y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
24	23	bieu 1014	. . . . . . . . . . . 12 ⊢ (∃!y(⟨A, B⟩ ∈ F ∧ y = B) ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
25		df-reu 1207	. . . . . . . . . . . 12 ⊢ (∃!y ∈ {B}⟨A, y⟩ ∈ F ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
26	24, 25	bitr4 154	. . . . . . . . . . 11 ⊢ (∃!y(⟨A, B⟩ ∈ F ∧ y = B) ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
27	17, 26	bitr3 153	. . . . . . . . . 10 ⊢ ((⟨A, B⟩ ∈ F ∧ ∃!y y = B) ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
28	16, 27	bitr 151	. . . . . . . . 9 ⊢ (⟨A, B⟩ ∈ F ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
29	13, 28	sylibr 175	. . . . . . . 8 ⊢ (F:{A}–→{B} → ⟨A, B⟩ ∈ F)
30	9, 29	syl5bir 184	. . . . . . 7 ⊢ ((x = A ∧ y = B) → (F:{A}–→{B} → ⟨x, y⟩ ∈ F))
31	30	com12 13	. . . . . 6 ⊢ (F:{A}–→{B} → ((x = A ∧ y = B) → ⟨x, y⟩ ∈ F))
32	7, 31	impbid 397	. . . . 5 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ (x = A ∧ y = B)))
33		opex 1893	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
34	33	elsnc 1826	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
35		visset 1350	. . . . . . 7 ⊢ x ∈ V
36	35, 1, 14	opth 1898	. . . . . 6 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
37	34, 36	bitr2 152	. . . . 5 ⊢ ((x = A ∧ y = B) ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
38	32, 37	syl6bb 414	. . . 4 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
39	38	19.21aivv 944	. . 3 ⊢ (F:{A}–→{B} → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
40		frel 2755	. . . . 5 ⊢ (F:{A}–→{B} → Rel F)
41	10	relsn 2485	. . . . 5 ⊢ Rel {⟨A, B⟩}
42	40, 41	jctir 241	. . . 4 ⊢ (F:{A}–→{B} → (Rel F ∧ Rel {⟨A, B⟩}))
43		cleqrel 2483	. . . 4 ⊢ ((Rel F ∧ Rel {⟨A, B⟩}) → (F = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
44	42, 43	syl 12	. . 3 ⊢ (F:{A}–→{B} → (F = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
45	39, 44	mpbird 171	. 2 ⊢ (F:{A}–→{B} → F = {⟨A, B⟩})
46	10, 14	f1osn 2827	. . . 4 ⊢ {⟨A, B⟩}:{A}–1-1-onto→{B}
47		f1oeq1 2795	. . . 4 ⊢ (F = {⟨A, B⟩} → (F:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
48	46, 47	mpbiri 169	. . 3 ⊢ (F = {⟨A, B⟩} → F:{A}–1-1-onto→{B})
49		f1of 2800	. . 3 ⊢ (F:{A}–1-1-onto→{B} → F:{A}–→{B})
50	48, 49	syl 12	. 2 ⊢ (F = {⟨A, B⟩} → F:{A}–→{B})
51	45, 50	impbi 139	1 ⊢ (F:{A}–→{B} ↔ F = {⟨A, B⟩})

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203  Vcvv 1348  {csn 1808  ⟨cop 1810  Rel wrel 2415  –→wf 2418  –1-1-onto→wf1o 2421

This theorem is referenced by:  fsn2 2896  mapsn 3269

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-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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