Theorem fsn 5278

Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

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

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelf 5005	. . . . . . . 8 ⊢ ((𝐹:{A}⟶{B} ∧ ⟨x, y⟩ ∈ 𝐹) → (x ∈ {A} ∧ y ∈ {B}))
2		elsn 3382	. . . . . . . . 9 ⊢ (x ∈ {A} ↔ x = A)
3		elsn 3382	. . . . . . . . 9 ⊢ (y ∈ {B} ↔ y = B)
4	2, 3	anbi12i 433	. . . . . . . 8 ⊢ ((x ∈ {A} ∧ y ∈ {B}) ↔ (x = A ∧ y = B))
5	1, 4	sylib 127	. . . . . . 7 ⊢ ((𝐹:{A}⟶{B} ∧ ⟨x, y⟩ ∈ 𝐹) → (x = A ∧ y = B))
6	5	ex 108	. . . . . 6 ⊢ (𝐹:{A}⟶{B} → (⟨x, y⟩ ∈ 𝐹 → (x = A ∧ y = B)))
7		fsn.1	. . . . . . . . . 10 ⊢ A ∈ V
8	7	snid 3394	. . . . . . . . 9 ⊢ A ∈ {A}
9		feu 5015	. . . . . . . . 9 ⊢ ((𝐹:{A}⟶{B} ∧ A ∈ {A}) → ∃!y ∈ {B}⟨A, y⟩ ∈ 𝐹)
10	8, 9	mpan2 401	. . . . . . . 8 ⊢ (𝐹:{A}⟶{B} → ∃!y ∈ {B}⟨A, y⟩ ∈ 𝐹)
11	3	anbi1i 431	. . . . . . . . . . 11 ⊢ ((y ∈ {B} ∧ ⟨A, y⟩ ∈ 𝐹) ↔ (y = B ∧ ⟨A, y⟩ ∈ 𝐹))
12		opeq2 3541	. . . . . . . . . . . . . 14 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
13	12	eleq1d 2103	. . . . . . . . . . . . 13 ⊢ (y = B → (⟨A, y⟩ ∈ 𝐹 ↔ ⟨A, B⟩ ∈ 𝐹))
14	13	pm5.32i 427	. . . . . . . . . . . 12 ⊢ ((y = B ∧ ⟨A, y⟩ ∈ 𝐹) ↔ (y = B ∧ ⟨A, B⟩ ∈ 𝐹))
15		ancom 253	. . . . . . . . . . . 12 ⊢ ((⟨A, B⟩ ∈ 𝐹 ∧ y = B) ↔ (y = B ∧ ⟨A, B⟩ ∈ 𝐹))
16	14, 15	bitr4i 176	. . . . . . . . . . 11 ⊢ ((y = B ∧ ⟨A, y⟩ ∈ 𝐹) ↔ (⟨A, B⟩ ∈ 𝐹 ∧ y = B))
17	11, 16	bitr2i 174	. . . . . . . . . 10 ⊢ ((⟨A, B⟩ ∈ 𝐹 ∧ y = B) ↔ (y ∈ {B} ∧ ⟨A, y⟩ ∈ 𝐹))
18	17	eubii 1906	. . . . . . . . 9 ⊢ (∃!y(⟨A, B⟩ ∈ 𝐹 ∧ y = B) ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ 𝐹))
19		fsn.2	. . . . . . . . . . . 12 ⊢ B ∈ V
20	19	eueq1 2707	. . . . . . . . . . 11 ⊢ ∃!y y = B
21	20	biantru 286	. . . . . . . . . 10 ⊢ (⟨A, B⟩ ∈ 𝐹 ↔ (⟨A, B⟩ ∈ 𝐹 ∧ ∃!y y = B))
22		euanv 1954	. . . . . . . . . 10 ⊢ (∃!y(⟨A, B⟩ ∈ 𝐹 ∧ y = B) ↔ (⟨A, B⟩ ∈ 𝐹 ∧ ∃!y y = B))
23	21, 22	bitr4i 176	. . . . . . . . 9 ⊢ (⟨A, B⟩ ∈ 𝐹 ↔ ∃!y(⟨A, B⟩ ∈ 𝐹 ∧ y = B))
24		df-reu 2307	. . . . . . . . 9 ⊢ (∃!y ∈ {B}⟨A, y⟩ ∈ 𝐹 ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ 𝐹))
25	18, 23, 24	3bitr4i 201	. . . . . . . 8 ⊢ (⟨A, B⟩ ∈ 𝐹 ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ 𝐹)
26	10, 25	sylibr 137	. . . . . . 7 ⊢ (𝐹:{A}⟶{B} → ⟨A, B⟩ ∈ 𝐹)
27		opeq12 3542	. . . . . . . 8 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
28	27	eleq1d 2103	. . . . . . 7 ⊢ ((x = A ∧ y = B) → (⟨x, y⟩ ∈ 𝐹 ↔ ⟨A, B⟩ ∈ 𝐹))
29	26, 28	syl5ibrcom 146	. . . . . 6 ⊢ (𝐹:{A}⟶{B} → ((x = A ∧ y = B) → ⟨x, y⟩ ∈ 𝐹))
30	6, 29	impbid 120	. . . . 5 ⊢ (𝐹:{A}⟶{B} → (⟨x, y⟩ ∈ 𝐹 ↔ (x = A ∧ y = B)))
31		vex 2554	. . . . . . . 8 ⊢ x ∈ V
32		vex 2554	. . . . . . . 8 ⊢ y ∈ V
33	31, 32	opex 3957	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
34	33	elsnc 3390	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
35	7, 19	opth2 3968	. . . . . 6 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
36	34, 35	bitr2i 174	. . . . 5 ⊢ ((x = A ∧ y = B) ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
37	30, 36	syl6bb 185	. . . 4 ⊢ (𝐹:{A}⟶{B} → (⟨x, y⟩ ∈ 𝐹 ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
38	37	alrimivv 1752	. . 3 ⊢ (𝐹:{A}⟶{B} → ∀x∀y(⟨x, y⟩ ∈ 𝐹 ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
39		frel 4992	. . . 4 ⊢ (𝐹:{A}⟶{B} → Rel 𝐹)
40	7, 19	relsnop 4387	. . . 4 ⊢ Rel {⟨A, B⟩}
41		eqrel 4372	. . . 4 ⊢ ((Rel 𝐹 ∧ Rel {⟨A, B⟩}) → (𝐹 = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ 𝐹 ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
42	39, 40, 41	sylancl 392	. . 3 ⊢ (𝐹:{A}⟶{B} → (𝐹 = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ 𝐹 ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
43	38, 42	mpbird 156	. 2 ⊢ (𝐹:{A}⟶{B} → 𝐹 = {⟨A, B⟩})
44	7, 19	f1osn 5109	. . . 4 ⊢ {⟨A, B⟩}:{A}–1-1-onto→{B}
45		f1oeq1 5060	. . . 4 ⊢ (𝐹 = {⟨A, B⟩} → (𝐹:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
46	44, 45	mpbiri 157	. . 3 ⊢ (𝐹 = {⟨A, B⟩} → 𝐹:{A}–1-1-onto→{B})
47		f1of 5069	. . 3 ⊢ (𝐹:{A}–1-1-onto→{B} → 𝐹:{A}⟶{B})
48	46, 47	syl 14	. 2 ⊢ (𝐹 = {⟨A, B⟩} → 𝐹:{A}⟶{B})
49	43, 48	impbii 117	1 ⊢ (𝐹:{A}⟶{B} ↔ 𝐹 = {⟨A, B⟩})

Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  ∃!wreu 2302  Vcvv 2551  {csn 3367  ⟨cop 3370  Rel wrel 4293  ⟶wf 4841  –1-1-onto→wf1o 4844

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  fsng  5279