Theorem fnsnb 7020

Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)

Hypothesis

Ref	Expression
fnsnb.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
fnsnb	⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Proof of Theorem fnsnb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnr 7019	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2		df-fn 6421	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3		fnsnb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	snid 4594	. . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴}
5		eleq2 2827	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴}))
6	4, 5	mpbiri 257	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
7	6	anim2i 616	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
8	2, 7	sylbi 216	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
9		funfvop 6909	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	8, 9	syl 17	. . . . . 6 ⊢ (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2826	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 246	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	1, 12	impbid 211	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4574	. . . 4 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	bitr4di 288	. . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
16	15	eqrdv 2736	. 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
17		fvex 6769	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
18	3, 17	fnsn 6476	. . 3 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}
19		fneq1 6508	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
20	18, 19	mpbiri 257	. 2 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴})
21	16, 20	impbii 208	1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564  dom cdm 5580  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  fnprb  7066