Theorem fnsnbg 7115

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 7114	. . . . . . 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 6592	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
4		snidg 4599	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6		fndm 6595	. . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
7	6	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
8	5, 7	eleqtrrd 2843	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9		funfvop 6998	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	3, 8, 9	syl2an2 692	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2828	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 248	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	2, 12	impbid 213	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4578	. . . . 5 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	bitr4di 290	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
16	15	eqrdv 2738	. . 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 6847	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
19		fnsng 6544	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
20	18, 19	mpan2 697	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
21		fneq1 6583	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
22	20, 21	syl5ibrcom 248	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴}))
23	17, 22	impbid 213	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562  ⟨cop 4568  dom cdm 5625  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  fnsnb  7116  frlmsnic  43027  termcnatval  50026