Theorem funsn 2690

Description: A singleton of an ordered pair is a function.

Hypotheses

Ref	Expression
funsn.1	⊢ A ∈ V
funsn.2	⊢ B ∈ V

Assertion

Ref	Expression
funsn	⊢ Fun {⟨A, B⟩}

Proof of Theorem funsn

Step	Hyp	Ref	Expression
1		funsn.1	. . . 4 ⊢ A ∈ V
2	1	relsn 2485	. . 3 ⊢ Rel {⟨A, B⟩}
3		cleq2 1110	. . . . . . 7 ⊢ (z = B → (y = z ↔ y = B))
4	3	biimparc 327	. . . . . 6 ⊢ ((y = B ∧ z = B) → y = z)
5		opex 1893	. . . . . . . 8 ⊢ ⟨x, y⟩ ∈ V
6	5	elsnc 1826	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
7		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
8		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
9		funsn.2	. . . . . . . . 9 ⊢ B ∈ V
10	7, 8, 9	opth 1898	. . . . . . . 8 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
11	10	pm3.27bd 263	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → y = B)
12	6, 11	sylbi 174	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} → y = B)
13		opex 1893	. . . . . . . 8 ⊢ ⟨x, z⟩ ∈ V
14	13	elsnc 1826	. . . . . . 7 ⊢ (⟨x, z⟩ ∈ {⟨A, B⟩} ↔ ⟨x, z⟩ = ⟨A, B⟩)
15		visset 1350	. . . . . . . . 9 ⊢ z ∈ V
16	7, 15, 9	opth 1898	. . . . . . . 8 ⊢ (⟨x, z⟩ = ⟨A, B⟩ ↔ (x = A ∧ z = B))
17	16	pm3.27bd 263	. . . . . . 7 ⊢ (⟨x, z⟩ = ⟨A, B⟩ → z = B)
18	14, 17	sylbi 174	. . . . . 6 ⊢ (⟨x, z⟩ ∈ {⟨A, B⟩} → z = B)
19	4, 12, 18	syl2an 349	. . . . 5 ⊢ ((⟨x, y⟩ ∈ {⟨A, B⟩} ∧ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
20	19	ax-gen 677	. . . 4 ⊢ ∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ∧ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
21	20	gen2 681	. . 3 ⊢ ∀x∀y∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ∧ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
22	2, 21	pm3.2i 234	. 2 ⊢ (Rel {⟨A, B⟩} ∧ ∀x∀y∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ∧ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z))
23		dffun4 2676	. 2 ⊢ (Fun {⟨A, B⟩} ↔ (Rel {⟨A, B⟩} ∧ ∀x∀y∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ∧ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)))
24	22, 23	mpbir 165	1 ⊢ Fun {⟨A, B⟩}

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  fun0 2691  f1osn 2827  fvsn 2879  tfrlem10 2958

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-fun 2432

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