Theorem fsn 5433

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 5236	. . . . . . 7 ⊢ ((F:{A}–→{B} ∧ ⟨x, y⟩ ∈ F) → (x ∈ {A} ∧ y ∈ {B}))
2		elsn 3749	. . . . . . . 8 ⊢ (x ∈ {A} ↔ x = A)
3		elsn 3749	. . . . . . . 8 ⊢ (y ∈ {B} ↔ y = B)
4	2, 3	anbi12i 678	. . . . . . 7 ⊢ ((x ∈ {A} ∧ y ∈ {B}) ↔ (x = A ∧ y = B))
5	1, 4	sylib 188	. . . . . 6 ⊢ ((F:{A}–→{B} ∧ ⟨x, y⟩ ∈ F) → (x = A ∧ y = B))
6	5	ex 423	. . . . 5 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F → (x = A ∧ y = B)))
7		fsn.1	. . . . . . . . 9 ⊢ A ∈ V
8	7	snid 3761	. . . . . . . 8 ⊢ A ∈ {A}
9		feu 5243	. . . . . . . 8 ⊢ ((F:{A}–→{B} ∧ A ∈ {A}) → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
10	8, 9	mpan2 652	. . . . . . 7 ⊢ (F:{A}–→{B} → ∃!y ∈ {B}⟨A, y⟩ ∈ F)
11	3	anbi1i 676	. . . . . . . . . 10 ⊢ ((y ∈ {B} ∧ ⟨A, y⟩ ∈ F) ↔ (y = B ∧ ⟨A, y⟩ ∈ F))
12		opeq2 4580	. . . . . . . . . . . . 13 ⊢ (y = B → ⟨A, y⟩ = ⟨A, B⟩)
13	12	eleq1d 2419	. . . . . . . . . . . 12 ⊢ (y = B → (⟨A, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
14	13	pm5.32i 618	. . . . . . . . . . 11 ⊢ ((y = B ∧ ⟨A, y⟩ ∈ F) ↔ (y = B ∧ ⟨A, B⟩ ∈ F))
15		ancom 437	. . . . . . . . . . 11 ⊢ ((⟨A, B⟩ ∈ F ∧ y = B) ↔ (y = B ∧ ⟨A, B⟩ ∈ F))
16	14, 15	bitr4i 243	. . . . . . . . . 10 ⊢ ((y = B ∧ ⟨A, y⟩ ∈ F) ↔ (⟨A, B⟩ ∈ F ∧ y = B))
17	11, 16	bitr2i 241	. . . . . . . . 9 ⊢ ((⟨A, B⟩ ∈ F ∧ y = B) ↔ (y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
18	17	eubii 2213	. . . . . . . 8 ⊢ (∃!y(⟨A, B⟩ ∈ F ∧ y = B) ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
19		fsn.2	. . . . . . . . . . 11 ⊢ B ∈ V
20	19	eueq1 3010	. . . . . . . . . 10 ⊢ ∃!y y = B
21	20	biantru 491	. . . . . . . . 9 ⊢ (⟨A, B⟩ ∈ F ↔ (⟨A, B⟩ ∈ F ∧ ∃!y y = B))
22		euanv 2265	. . . . . . . . 9 ⊢ (∃!y(⟨A, B⟩ ∈ F ∧ y = B) ↔ (⟨A, B⟩ ∈ F ∧ ∃!y y = B))
23	21, 22	bitr4i 243	. . . . . . . 8 ⊢ (⟨A, B⟩ ∈ F ↔ ∃!y(⟨A, B⟩ ∈ F ∧ y = B))
24		df-reu 2622	. . . . . . . 8 ⊢ (∃!y ∈ {B}⟨A, y⟩ ∈ F ↔ ∃!y(y ∈ {B} ∧ ⟨A, y⟩ ∈ F))
25	18, 23, 24	3bitr4i 268	. . . . . . 7 ⊢ (⟨A, B⟩ ∈ F ↔ ∃!y ∈ {B}⟨A, y⟩ ∈ F)
26	10, 25	sylibr 203	. . . . . 6 ⊢ (F:{A}–→{B} → ⟨A, B⟩ ∈ F)
27		opeq12 4581	. . . . . . 7 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
28	27	eleq1d 2419	. . . . . 6 ⊢ ((x = A ∧ y = B) → (⟨x, y⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
29	26, 28	syl5ibrcom 213	. . . . 5 ⊢ (F:{A}–→{B} → ((x = A ∧ y = B) → ⟨x, y⟩ ∈ F))
30	6, 29	impbid 183	. . . 4 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ (x = A ∧ y = B)))
31		vex 2863	. . . . . . 7 ⊢ x ∈ V
32		vex 2863	. . . . . . 7 ⊢ y ∈ V
33	31, 32	opex 4589	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
34	33	elsnc 3757	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
35		opth 4603	. . . . 5 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
36	34, 35	bitr2i 241	. . . 4 ⊢ ((x = A ∧ y = B) ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
37	30, 36	syl6bb 252	. . 3 ⊢ (F:{A}–→{B} → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨A, B⟩}))
38	37	eqrelrdv 4853	. 2 ⊢ (F:{A}–→{B} → F = {⟨A, B⟩})
39	7, 19	f1osn 5323	. . . 4 ⊢ {⟨A, B⟩}:{A}–1-1-onto→{B}
40		f1oeq1 5282	. . . 4 ⊢ (F = {⟨A, B⟩} → (F:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
41	39, 40	mpbiri 224	. . 3 ⊢ (F = {⟨A, B⟩} → F:{A}–1-1-onto→{B})
42		f1of 5288	. . 3 ⊢ (F:{A}–1-1-onto→{B} → F:{A}–→{B})
43	41, 42	syl 15	. 2 ⊢ (F = {⟨A, B⟩} → F:{A}–→{B})
44	38, 43	impbii 180	1 ⊢ (F:{A}–→{B} ↔ F = {⟨A, B⟩})

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃!wreu 2617  Vcvv 2860  {csn 3738  ⟨cop 4562  –→wf 4778  –1-1-onto→wf1o 4781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is used by:  fsng  5434  fsn2  5435  xpsn  5436  mapsn  6027