Theorem fnsnbt 40208

Description: A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)

Assertion

Ref	Expression
fnsnbt	⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))

Proof of Theorem fnsnbt

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnr 7037	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2	1	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
3		fnfun 6533	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
4		snidg 4595	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6		fndm 6536	. . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
7	6	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
8	5, 7	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9		funfvop 6927	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	3, 8, 9	syl2an2 683	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 246	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	2, 12	impbid 211	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4577	. . . . 5 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	bitr4di 289	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
16	15	eqrdv 2736	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
17	16	ex 413	. 2 ⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
18		fvex 6787	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
19		fnsng 6486	. . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
20	18, 19	mpan2 688	. . 3 ⊢ (𝐴 ∈ V → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
21		fneq1 6524	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
22	20, 21	syl5ibrcom 246	. 2 ⊢ (𝐴 ∈ V → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴}))
23	17, 22	impbid 211	1 ⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567  dom cdm 5589  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  frlmsnic  40263