Theorem fsn 5737

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 5432	. . . . . . . 8 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}))
2		velsn 3640	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 3640	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
4	2, 3	anbi12i 460	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	sylib 122	. . . . . . 7 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6	5	ex 115	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
7		fsn.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
8	7	snid 3654	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴}
9		feu 5443	. . . . . . . . 9 ⊢ ((𝐹:{𝐴}⟶{𝐵} ∧ 𝐴 ∈ {𝐴}) → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
10	8, 9	mpan2 425	. . . . . . . 8 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
11	3	anbi1i 458	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
12		opeq2 3810	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
13	12	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
14	13	pm5.32i 454	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
15		ancom 266	. . . . . . . . . . . 12 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
16	14, 15	bitr4i 187	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
17	11, 16	bitr2i 185	. . . . . . . . . 10 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
18	17	eubii 2054	. . . . . . . . 9 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
19		fsn.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
20	19	eueq1 2936	. . . . . . . . . . 11 ⊢ ∃!𝑦 𝑦 = 𝐵
21	20	biantru 302	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
22		euanv 2102	. . . . . . . . . 10 ⊢ (∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ ∃!𝑦 𝑦 = 𝐵))
23	21, 22	bitr4i 187	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦(⟨𝐴, 𝐵⟩ ∈ 𝐹 ∧ 𝑦 = 𝐵))
24		df-reu 2482	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
25	18, 23, 24	3bitr4i 212	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ∃!𝑦 ∈ {𝐵}⟨𝐴, 𝑦⟩ ∈ 𝐹)
26	10, 25	sylibr 134	. . . . . . 7 ⊢ (𝐹:{𝐴}⟶{𝐵} → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
27		opeq12 3811	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	eleq1d 2265	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
29	26, 28	syl5ibrcom 157	. . . . . 6 ⊢ (𝐹:{𝐴}⟶{𝐵} → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝐹))
30	6, 29	impbid 129	. . . . 5 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
31		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
32		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
33	31, 32	opex 4263	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
34	33	elsn 3639	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
35	7, 19	opth2 4274	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
36	34, 35	bitr2i 185	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
37	30, 36	bitrdi 196	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
38	37	alrimivv 1889	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}))
39		frel 5415	. . . 4 ⊢ (𝐹:{𝐴}⟶{𝐵} → Rel 𝐹)
40	7, 19	relsnop 4770	. . . 4 ⊢ Rel {⟨𝐴, 𝐵⟩}
41		eqrel 4753	. . . 4 ⊢ ((Rel 𝐹 ∧ Rel {⟨𝐴, 𝐵⟩}) → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
42	39, 40, 41	sylancl 413	. . 3 ⊢ (𝐹:{𝐴}⟶{𝐵} → (𝐹 = {⟨𝐴, 𝐵⟩} ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})))
43	38, 42	mpbird 167	. 2 ⊢ (𝐹:{𝐴}⟶{𝐵} → 𝐹 = {⟨𝐴, 𝐵⟩})
44	7, 19	f1osn 5547	. . . 4 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
45		f1oeq1 5495	. . . 4 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → (𝐹:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
46	44, 45	mpbiri 168	. . 3 ⊢ (𝐹 = {⟨𝐴, 𝐵⟩} → 𝐹:{𝐴}–1-1-onto→{𝐵})
47		f1of 5507	. . 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 126	1 ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃!weu 2045   ∈ wcel 2167  ∃!wreu 2477  Vcvv 2763  {csn 3623  ⟨cop 3626  Rel wrel 4669  ⟶wf 5255  –1-1-onto→wf1o 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266

This theorem is referenced by:  fsng  5738  mapsn  6758