Theorem fsn 5657

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 5359	. . . . . . . 8 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2		velsn 3593	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 3593	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
4	2, 3	anbi12i 456	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	sylib 121	. . . . . . 7 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6	5	ex 114	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
7		fsn.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8	7	snid 3607	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴}
9		feu 5370	. . . . . . . . 9 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
10	8, 9	mpan2 422	. . . . . . . 8 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
11	3	anbi1i 454	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12		opeq2 3759	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
13	12	eleq1d 2235	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
14	13	pm5.32i 450	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15		ancom 264	. . . . . . . . . . . 12 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
16	14, 15	bitr4i 186	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
17	11, 16	bitr2i 184	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
18	17	eubii 2023	. . . . . . . . 9 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
19		fsn.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
20	19	eueq1 2898	. . . . . . . . . . 11 ⊢ ∃!𝑦 𝑦 = 𝐵
21	20	biantru 300	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22		euanv 2071	. . . . . . . . . 10 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
23	21, 22	bitr4i 186	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
24		df-reu 2451	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
25	18, 23, 24	3bitr4i 211	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
26	10, 25	sylibr 133	. . . . . . 7 ⊢ (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
27		opeq12 3760	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	eleq1d 2235	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
29	26, 28	syl5ibrcom 156	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
30	6, 29	impbid 128	. . . . 5 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
31		vex 2729	. . . . . . . 8 ⊢ 𝑥 ∈ V
32		vex 2729	. . . . . . . 8 ⊢ 𝑦 ∈ V
33	31, 32	opex 4207	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
34	33	elsn 3592	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
35	7, 19	opth2 4218	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
36	34, 35	bitr2i 184	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
37	30, 36	bitrdi 195	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
38	37	alrimivv 1863	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
39		frel 5342	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
40	7, 19	relsnop 4710	. . . 4 ⊢ Rel {⟨𝐴, 𝐵⟩}
41		eqrel 4693	. . . 4 ⊢ ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
42	39, 40, 41	sylancl 410	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
43	38, 42	mpbird 166	. 2 ⊢ (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
44	7, 19	f1osn 5472	. . . 4 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
45		f1oeq1 5421	. . . 4 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
46	44, 45	mpbiri 167	. . 3 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
47		f1of 5432	. . 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 125	1 ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃!weu 2014   ∈ wcel 2136  ∃!wreu 2446  Vcvv 2726  {csn 3576  ⟨cop 3579  Rel wrel 4609  ⟶wf 5184  –1-1-onto→wf1o 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  fsng  5658  mapsn  6656