Theorem fsn 6907

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 6539	. . . . . . . 8 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2		velsn 4532	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 4532	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
4	2, 3	anbi12i 630	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	sylib 221	. . . . . . 7 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6	5	ex 416	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
7		fsn.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8	7	snid 4552	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴}
9		feu 6554	. . . . . . . . 9 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
10	8, 9	mpan2 691	. . . . . . . 8 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
11	3	anbi1i 627	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12		opeq2 4761	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
13	12	eleq1d 2817	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
14	13	pm5.32i 578	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15	14	biancomi 466	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
16	11, 15	bitr2i 279	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
17	16	eubii 2586	. . . . . . . . 9 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
18		fsn.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
19	18	eueqi 3608	. . . . . . . . . . 11 ⊢ ∃!𝑦 𝑦 = 𝐵
20	19	biantru 533	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
21		euanv 2627	. . . . . . . . . 10 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22	20, 21	bitr4i 281	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
23		df-reu 3060	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
24	17, 22, 23	3bitr4i 306	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
25	10, 24	sylibr 237	. . . . . . 7 ⊢ (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
26		opeq12 4763	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
27	26	eleq1d 2817	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
28	25, 27	syl5ibrcom 250	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
29	6, 28	impbid 215	. . . . 5 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
30		opex 5322	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30	elsn 4531	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32	7, 18	opth2 5338	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
33	31, 32	bitr2i 279	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
34	29, 33	bitrdi 290	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
35	34	alrimivv 1935	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
36		frel 6509	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
37	7, 18	relsnop 5649	. . . 4 ⊢ Rel {⟨𝐴, 𝐵⟩}
38		eqrel 5629	. . . 4 ⊢ ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
39	36, 37, 38	sylancl 589	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
40	35, 39	mpbird 260	. 2 ⊢ (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
41	7, 18	f1osn 6657	. . . 4 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
42		f1oeq1 6606	. . . 4 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
43	41, 42	mpbiri 261	. . 3 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
44		f1of 6618	. . 3 ⊢ (𝐹:{𝐴}–1-1-onto→{𝐵} → 𝐹:{𝐴}⟶{𝐵})
45	43, 44	syl 17	. 2 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}⟶{𝐵})
46	40, 45	impbii 212	1 ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  ∃!wreu 3055  Vcvv 3398  {csn 4516  ⟨cop 4522  Rel wrel 5530  ⟶wf 6335  –1-1-onto→wf1o 6338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-reu 3060  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346

This theorem is referenced by:  fsn2  6908  fsng  6909  axlowdimlem7  26894  poimirlem3  35403  poimirlem9  35409  fdc  35526