Theorem fnsnbt 41357

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 7164	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2	1	adantl 480	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
3		fnfun 6648	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
4		snidg 4661	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
5	4	adantr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6		fndm 6651	. . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
7	6	adantl 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
8	5, 7	eleqtrrd 2834	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9		funfvop 7050	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	3, 8, 9	syl2an2 682	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2819	. . . . . . 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 4643	. . . . 5 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	bitr4di 288	. . . 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 411	. 2 ⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
18		fvex 6903	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
19		fnsng 6599	. . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐹‘𝐴) ∈ V) → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
20	18, 19	mpan2 687	. . 3 ⊢ (𝐴 ∈ V → {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴})
21		fneq1 6639	. . 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472  {csn 4627  ⟨cop 4633  dom cdm 5675  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  frlmsnic  41412