Theorem fnsnbg 7093

Description: A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.) Relax condition for being in the universal class. (Revised by Zhi Wang, 21-Oct-2025.)

Assertion

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

Proof of Theorem fnsnbg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnr 7092	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2	1	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
3		fnfun 6577	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
4		snidg 4611	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6		fndm 6580	. . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
8	5, 7	eleqtrrd 2832	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9		funfvop 6978	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	3, 8, 9	syl2an2 686	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2817	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 247	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	2, 12	impbid 212	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4590	. . . . 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 2728	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
17	16	ex 412	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
18		fvex 6830	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
19		fnsng 6529	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
20	18, 19	mpan2 691	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
21		fneq1 6568	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
22	20, 21	syl5ibrcom 247	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴}))
23	17, 22	impbid 212	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Vcvv 3434  {csn 4574  ⟨cop 4580  dom cdm 5614  Fun wfun 6471   Fn wfn 6472  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485

This theorem is referenced by:  fnsnb  7094  frlmsnic  42552  termcnatval  49546