Theorem fnsnb 6753

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 6752	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2		df-fn 6193	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3		fnsnb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	snid 4474	. . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴}
5		eleq2 2854	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴}))
6	4, 5	mpbiri 250	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
7	6	anim2i 607	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
8	2, 7	sylbi 209	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
9		funfvop 6647	. . . . . . 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 2853	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 239	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	1, 12	impbid 204	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4458	. . . 4 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	syl6bbr 281	. . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
16	15	eqrdv 2776	. 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
17		fvex 6514	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
18	3, 17	fnsn 6247	. . 3 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}
19		fneq1 6279	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
20	18, 19	mpbiri 250	. 2 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴})
21	16, 20	impbii 201	1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3415  {csn 4442  ⟨cop 4448  dom cdm 5408  Fun wfun 6184   Fn wfn 6185  ‘cfv 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198

This theorem is referenced by:  fnprb  6799