Theorem funinsn 5386

Description: A function based on the singleton of an ordered pair. Unlike funsng 5383, this holds even if 𝐴 or 𝐵 is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)

Assertion

Ref	Expression
funinsn	⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Proof of Theorem funinsn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3430	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (𝑉 × 𝑊)
2		xpss 4840	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
3	1, 2	sstri 3237	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V)
4		df-rel 4738	. . 3 ⊢ (Rel ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V))
5	3, 4	mpbir 146	. 2 ⊢ Rel ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))
6		elin 3392	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑊)))
7	6	simplbi 274	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 3691	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
9	7, 8	syl 14	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
10		vex 2806	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 2806	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	opth 4335	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
13	9, 12	sylib 122	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
14	13	simprd 114	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → 𝑦 = 𝐵)
15		elin 3392	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑧⟩ ∈ (𝑉 × 𝑊)))
16	15	simplbi 274	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩})
17		elsni 3691	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑥, 𝑧⟩ = ⟨𝐴, 𝐵⟩)
18	16, 17	syl 14	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑧⟩ = ⟨𝐴, 𝐵⟩)
19		vex 2806	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	10, 19	opth 4335	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑧 = 𝐵))
21	18, 20	sylib 122	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → (𝑥 = 𝐴 ∧ 𝑧 = 𝐵))
22	21	simprd 114	. . . . 5 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → 𝑧 = 𝐵)
23		eqtr3 2251	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 = 𝑧)
24	14, 22, 23	syl2an 289	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
25	24	gen2 1499	. . 3 ⊢ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
26	25	ax-gen 1498	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
27		dffun4 5344	. 2 ⊢ (Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (Rel ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)))
28	5, 26, 27	mpbir2an 951	1 ⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∩ cin 3200   ⊆ wss 3201  {csn 3673  ⟨cop 3676   × cxp 4729  Rel wrel 4736  Fun wfun 5327

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-fun 5335

This theorem is referenced by: (None)