Theorem funinsn 5247

Description: A function based on the singleton of an ordered pair. Unlike funsng 5244, 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 3348	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (𝑉 × 𝑊)
2		xpss 4719	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
3	1, 2	sstri 3156	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V)
4		df-rel 4618	. . 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 3310	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑊)))
7	6	simplbi 272	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 3601	. . . . . . . 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 2733	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	opth 4222	. . . . . . 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 3310	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ↔ (⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩} ∧ ⟨𝑥, 𝑧⟩ ∈ (𝑉 × 𝑊)))
16	15	simplbi 272	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) → ⟨𝑥, 𝑧⟩ ∈ {⟨𝐴, 𝐵⟩})
17		elsni 3601	. . . . . . . 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 2733	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	10, 19	opth 4222	. . . . . . 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 2190	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 = 𝑧)
24	14, 22, 23	syl2an 287	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
25	24	gen2 1443	. . 3 ⊢ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
26	25	ax-gen 1442	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
27		dffun4 5209	. 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 937	1 ⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  {csn 3583  ⟨cop 3586   × cxp 4609  Rel wrel 4616  Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-fun 5200

This theorem is referenced by: (None)