Theorem funsn 3535

Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65.

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		dffun4 3520	. 2 ⊢ (Fun {⟨A, B⟩} ↔ (Rel {⟨A, B⟩} ⋀ ∀x∀y∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ⋀ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)))
2		funsn.1	. . 3 ⊢ A ∈ V
3	2	relsn 3249	. 2 ⊢ Rel {⟨A, B⟩}
4		eqtr3t 1491	. . . . 5 ⊢ ((y = B ⋀ z = B) → y = z)
5		opex 2777	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
6	5	elsnc 2427	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
7		visset 1809	. . . . . . 7 ⊢ y ∈ V
8		funsn.2	. . . . . . 7 ⊢ B ∈ V
9	7, 8	opth2 2795	. . . . . 6 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → y = B)
10	6, 9	sylbi 199	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} → y = B)
11		opex 2777	. . . . . . 7 ⊢ ⟨x, z⟩ ∈ V
12	11	elsnc 2427	. . . . . 6 ⊢ (⟨x, z⟩ ∈ {⟨A, B⟩} ↔ ⟨x, z⟩ = ⟨A, B⟩)
13		visset 1809	. . . . . . 7 ⊢ z ∈ V
14	13, 8	opth2 2795	. . . . . 6 ⊢ (⟨x, z⟩ = ⟨A, B⟩ → z = B)
15	12, 14	sylbi 199	. . . . 5 ⊢ (⟨x, z⟩ ∈ {⟨A, B⟩} → z = B)
16	4, 10, 15	syl2an 454	. . . 4 ⊢ ((⟨x, y⟩ ∈ {⟨A, B⟩} ⋀ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
17	16	ax-gen 961	. . 3 ⊢ ∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ⋀ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
18	17	gen2 981	. 2 ⊢ ∀x∀y∀z((⟨x, y⟩ ∈ {⟨A, B⟩} ⋀ ⟨x, z⟩ ∈ {⟨A, B⟩}) → y = z)
19	1, 3, 18	mpbir2an 729	1 ⊢ Fun {⟨A, B⟩}

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  Vcvv 1807  {csn 2405  ⟨cop 2407  Rel wrel 3170  Fun wfun 3171

This theorem is referenced by:  fun0 3536  f1osn 3710  fvsn 3785  tfrlem10 3911  ringsn 8115  1alg 10534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-fun 3187

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