Theorem fsn 6899

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 6541	. . . . . . . 8 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2		velsn 4585	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 4585	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
4	2, 3	anbi12i 628	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	sylib 220	. . . . . . 7 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6	5	ex 415	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
7		fsn.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8	7	snid 4603	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴}
9		feu 6556	. . . . . . . . 9 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
10	8, 9	mpan2 689	. . . . . . . 8 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
11	3	anbi1i 625	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12		opeq2 4806	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
13	12	eleq1d 2899	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
14	13	pm5.32i 577	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15	14	biancomi 465	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
16	11, 15	bitr2i 278	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
17	16	eubii 2670	. . . . . . . . 9 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
18		fsn.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
19	18	eueqi 3702	. . . . . . . . . . 11 ⊢ ∃!𝑦 𝑦 = 𝐵
20	19	biantru 532	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
21		euanv 2709	. . . . . . . . . 10 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22	20, 21	bitr4i 280	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
23		df-reu 3147	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
24	17, 22, 23	3bitr4i 305	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
25	10, 24	sylibr 236	. . . . . . 7 ⊢ (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
26		opeq12 4807	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
27	26	eleq1d 2899	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
28	25, 27	syl5ibrcom 249	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
29	6, 28	impbid 214	. . . . 5 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
30		opex 5358	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
31	30	elsn 4584	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32	7, 18	opth2 5374	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
33	31, 32	bitr2i 278	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
34	29, 33	syl6bb 289	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
35	34	alrimivv 1929	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
36		frel 6521	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
37	7, 18	relsnop 5680	. . . 4 ⊢ Rel {⟨𝐴, 𝐵⟩}
38		eqrel 5660	. . . 4 ⊢ ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
39	36, 37, 38	sylancl 588	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
40	35, 39	mpbird 259	. 2 ⊢ (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
41	7, 18	f1osn 6656	. . . 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 260	. . 3 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
44		f1of 6617	. . 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 211	1 ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃!weu 2653  ∃!wreu 3142  Vcvv 3496  {csn 4569  ⟨cop 4575  Rel wrel 5562  ⟶wf 6353  –1-1-onto→wf1o 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364

This theorem is referenced by:  fsn2  6900  fsng  6901  axlowdimlem7  26736  poimirlem3  34897  poimirlem9  34903  fdc  35022