Theorem funinsn 5108

Description: A function based on the singleton of an ordered pair. Unlike funsng 5105, 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 3244	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (𝑉 × 𝑊)
2		xpss 4585	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
3	1, 2	sstri 3056	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V)
4		df-rel 4484	. . 3 ⊢ (Rel ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V))
5	3, 4	mpbir 145	. 2 ⊢ Rel ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))
6		elin 3206	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑊)))
7	6	simplbi 270	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 3492	. . . . . . . 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 2644	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 2644	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	opth 4097	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
13	9, 12	sylib 121	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
14	13	simprd 113	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → 𝑦 = 𝐵)
15		elin 3206	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑧⟩ ∈ (𝑉 × 𝑊)))
16	15	simplbi 270	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩})
17		elsni 3492	. . . . . . . 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 2644	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	10, 19	opth 4097	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑧 = 𝐵))
21	18, 20	sylib 121	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → (𝑥 = 𝐴 ∧ 𝑧 = 𝐵))
22	21	simprd 113	. . . . 5 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → 𝑧 = 𝐵)
23		eqtr3 2119	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 = 𝑧)
24	14, 22, 23	syl2an 285	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
25	24	gen2 1394	. . 3 ⊢ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
26	25	ax-gen 1393	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
27		dffun4 5070	. 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 894	1 ⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1297   = wceq 1299   ∈ wcel 1448  Vcvv 2641   ∩ cin 3020   ⊆ wss 3021  {csn 3474  ⟨cop 3477   × cxp 4475  Rel wrel 4482  Fun wfun 5053

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-fun 5061

This theorem is referenced by: (None)