Theorem fsn 3825

Description: A function maps a singleton to a singleton iff it is the singleton of a ordered pair.

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

Step	Hyp	Ref	Expression
1		visset 1809	. . . . . . . . 9 ⊢ y ∈ V
2	1	opelf 3631	. . . . . . . 8 ⊢ ((F:{A}–→{B} ⋀ ⟨x, y⟩ ∈ F) → (x ∈ {A} ⋀ y ∈ {B}))
3		elsn 2417	. . . . . . . . 9 ⊢ (x ∈ {A} ↔ x = A)
4		elsn 2417	. . . . . . . . 9 ⊢ (y ∈ {B} ↔ y = B)
5	3, 4	anbi12i 482	. . . . . . . 8 ⊢ ((x ∈ {A} ⋀ y ∈ {B}) ↔ (x = A ⋀ y = B))
6	2, 5	sylib 198	. . . . . . 7 ⊢ ((F:{A}–→{B} ⋀ ⟨x, y⟩ ∈ F) → (x = A ⋀ y = B))
7	6	ex 373	. . . . . 6 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F → (x = A ⋀ y = B)))
8		opeq12 2485	. . . . . . . 8 ⊢ ((x = A ⋀ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
9	8	eleq1d 1537	. . . . . . 7 ⊢ ((x = A ⋀ y = B) → (⟨x, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
10		fsn.1	. . . . . . . . . 10 ⊢ A ∈ V
11	10	snid 2431	. . . . . . . . 9 ⊢ A ∈ {A}
12		feu 3638	. . . . . . . . 9 ⊢ ((F:{A}–→{B} ⋀ A ∈ {A}) → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
13	11, 12	mpan2 695	. . . . . . . 8 ⊢ (F:{A}–→{B} → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
14		fsn.2	. . . . . . . . . . 11 ⊢ B ∈ V
15	14	eueq1 1913	. . . . . . . . . 10 ⊢ ∃!y y = B
16	15	biantru 723	. . . . . . . . 9 ⊢ (⟨A, B⟩ ∈ F ↔ (⟨A, B⟩ ∈ F ⋀ ∃!y y = B))
17		euanv 1430	. . . . . . . . 9 ⊢ (∃!y(⟨A, B⟩ ∈ F ⋀ y = B) ↔ (⟨A, B⟩ ∈ F ⋀ ∃!y y = B))
18		opeq2 2484	. . . . . . . . . . . . . 14 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
19	18	eleq1d 1537	. . . . . . . . . . . . 13 ⊢ (y = B → (⟨A, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
20	19	pm5.32i 644	. . . . . . . . . . . 12 ⊢ ((y = B ⋀ ⟨A, y⟩ ∈ F) ↔ (y = B ⋀ ⟨A, B⟩ ∈ F))
21	4	anbi1i 481	. . . . . . . . . . . 12 ⊢ ((y ∈ {B} ⋀ ⟨A, y⟩ ∈ F) ↔ (y = B ⋀ ⟨A, y⟩ ∈ F))
22		ancom 435	. . . . . . . . . . . 12 ⊢ ((⟨A, B⟩ ∈ F ⋀ y = B) ↔ (y = B ⋀ ⟨A, B⟩ ∈ F))
23	20, 21, 22	3bitr4r 184	. . . . . . . . . . 11 ⊢ ((⟨A, B⟩ ∈ F ⋀ y = B) ↔ (y ∈ {B} ⋀ ⟨A, y⟩ ∈ F))
24	23	eubii 1385	. . . . . . . . . 10 ⊢ (∃!y(⟨A, B⟩ ∈ F ⋀ y = B) ↔ ∃!y(y ∈ {B} ⋀ ⟨A, y⟩ ∈ F))
25		df-reu 1648	. . . . . . . . . 10 ⊢ (∃!y ∈ {B}⟨A, y⟩ ∈ F ↔ ∃!y(y ∈ {B} ⋀ ⟨A, y⟩ ∈ F))
26	24, 25	bitr4 176	. . . . . . . . 9 ⊢ (∃!y(⟨A, B⟩ ∈ F ⋀ y = B) ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
27	16, 17, 26	3bitr2 179	. . . . . . . 8 ⊢ (⟨A, B⟩ ∈ F ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
28	13, 27	sylibr 200	. . . . . . 7 ⊢ (F:{A}–→{B} → ⟨A, B⟩ ∈ F)
29	9, 28	syl5cbir 211	. . . . . 6 ⊢ (F:{A}–→{B} → ((x = A ⋀ y = B) → ⟨x, y⟩ ∈ F))
30	7, 29	impbid 515	. . . . 5 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ (x = A ⋀ y = B)))
31		opex 2777	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
32	31	elsnc 2427	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
33		visset 1809	. . . . . . 7 ⊢ x ∈ V
34	33, 1, 14	opth 2782	. . . . . 6 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ⋀ y = B))
35	32, 34	bitr2 174	. . . . 5 ⊢ ((x = A ⋀ y = B) ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
36	30, 35	syl6bb 535	. . . 4 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
37	36	19.21aivv 1285	. . 3 ⊢ (F:{A}–→{B} → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
38		frel 3622	. . . . 5 ⊢ (F:{A}–→{B} → Rel F)
39	10	relsn 3249	. . . . 5 ⊢ Rel {⟨A, B⟩}
40	38, 39	jctir 293	. . . 4 ⊢ (F:{A}–→{B} → (Rel F ⋀ Rel {⟨A, B⟩}))
41		eqrel 3245	. . . 4 ⊢ ((Rel F ⋀ Rel {⟨A, B⟩}) → (F = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
42	40, 41	syl 10	. . 3 ⊢ (F:{A}–→{B} → (F = {⟨A, B⟩} ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})))
43	37, 42	mpbird 196	. 2 ⊢ (F:{A}–→{B} → F = {⟨A, B⟩})
44	10, 14	f1osn 3710	. . . 4 ⊢ {⟨A, B⟩}:{A}–1-1-onto→{B}
45		f1oeq1 3675	. . . 4 ⊢ (F = {⟨A, B⟩} → (F:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
46	44, 45	mpbiri 194	. . 3 ⊢ (F = {⟨A, B⟩} → F:{A}–1-1-onto→{B})
47		f1of 3680	. . 3 ⊢ (F:{A}–1-1-onto→{B} → F:{A}–→{B})
48	46, 47	syl 10	. 2 ⊢ (F = {⟨A, B⟩} → F:{A}–→{B})
49	43, 48	impbi 157	1 ⊢ (F:{A}–→{B} ↔ F = {⟨A, B⟩})

Syntax hints:   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃!weu 1378  ∃!wreu 1644  Vcvv 1807  {csn 2405  ⟨cop 2407  Rel wrel 3170  –→wf 3173  –1-1-onto→wf1o 3176

This theorem is referenced by:  xpsn 3826  fsn2 3827  mapsn 4335

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-reu 1648  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192

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