Theorem funinsn 5172

Description: A function based on the singleton of an ordered pair. Unlike funsng 5169, 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 3297	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (𝑉 × 𝑊)
2		xpss 4647	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
3	1, 2	sstri 3106	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ⊆ (V × V)
4		df-rel 4546	. . 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 3259	. . . . . . . . 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 3545	. . . . . . . 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 2689	. . . . . . . 8 ⊢ 𝑥 ∈ V
11		vex 2689	. . . . . . . 8 ⊢ 𝑦 ∈ V
12	10, 11	opth 4159	. . . . . . 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 3259	. . . . . . . . 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 3545	. . . . . . . 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 2689	. . . . . . . 8 ⊢ 𝑧 ∈ V
20	10, 19	opth 4159	. . . . . . 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 2159	. . . . 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 1426	. . 3 ⊢ ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
26	25	ax-gen 1425	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊)) ∧ ⟨𝑥, 𝑧⟩ ∈ ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))) → 𝑦 = 𝑧)
27		dffun4 5134	. 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 926	1 ⊢ Fun ({⟨𝐴, 𝐵⟩} ∩ (𝑉 × 𝑊))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071  {csn 3527  ⟨cop 3530   × cxp 4537  Rel wrel 4544  Fun wfun 5117

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125

This theorem is referenced by: (None)