Theorem funinsn 5307

Description: A function based on the singleton of an ordered pair. Unlike funsng 5304, 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 3384	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (𝑉 × 𝑊)
2		xpss 4771	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
3	1, 2	sstri 3192	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V)
4		df-rel 4670	. . 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 3346	. . . . . . . . 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 3640	. . . . . . . 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 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	opth 4270	. . . . . . 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 3346	. . . . . . . . 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 3640	. . . . . . . 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 2766	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	10, 19	opth 4270	. . . . . . 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 2216	. . . . 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 1464	. . 3 ⊢ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
26	25	ax-gen 1463	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
27		dffun4 5269	. 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 944	1 ⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  {csn 3622  ⟨cop 3625   × cxp 4661  Rel wrel 4668  Fun wfun 5252

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-fun 5260

This theorem is referenced by: (None)