Theorem fnsnbg 7153

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 7152	. . . . . . 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 6635	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
4		snidg 4634	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6		fndm 6638	. . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
8	5, 7	eleqtrrd 2836	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9		funfvop 7037	. . . . . . . 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 2821	. . . . . . 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 4615	. . . . 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 2732	. . 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 6886	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
19		fnsng 6585	. . . 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 6626	. . 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 1539   ∈ wcel 2107  Vcvv 3457  {csn 4599  ⟨cop 4605  dom cdm 5652  Fun wfun 6522   Fn wfn 6523  ‘cfv 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536

This theorem is referenced by:  fnsnb  7154  frlmsnic  42495  termcnatval  49281