Theorem fnsnbOLD 7166

Description: Obsolete version of fnsnb 7165 as of 21-Oct-2025. 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.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
fnsnb.1	⊢ 𝐴 ∈ V

Assertion

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

Proof of Theorem fnsnbOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnr 7163	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 → 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
2		df-fn 6541	. . . . . . . 8 ⊢ (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3		fnsnb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	snid 4629	. . . . . . . . . 10 ⊢ 𝐴 ∈ {𝐴}
5		eleq2 2852	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ {𝐴}))
6	4, 5	mpbiri 261	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
7	6	anim2i 628	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
8	2, 7	sylbi 220	. . . . . . 7 ⊢ (𝐹 Fn {𝐴} → (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))
9		funfvop 7047	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
10	8, 9	syl 18	. . . . . 6 ⊢ (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
11		eleq1 2851	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → (𝑥 ∈ 𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
12	10, 11	syl5ibrcom 250	. . . . 5 ⊢ (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩ → 𝑥 ∈ 𝐹))
13	1, 12	impbid 215	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩))
14		velsn 4606	. . . 4 ⊢ (𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹‘𝐴)⟩)
15	13, 14	bitr4di 292	. . 3 ⊢ (𝐹 Fn {𝐴} → (𝑥 ∈ 𝐹 ↔ 𝑥 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
16	15	eqrdv 2761	. 2 ⊢ (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
17		fvex 6896	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
18	3, 17	fnsn 6596	. . 3 ⊢ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}
19		fneq1 6628	. . 3 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹‘𝐴)⟩} Fn {𝐴}))
20	18, 19	mpbiri 261	. 2 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐹 Fn {𝐴})
21	16, 20	impbii 212	1 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4590  ⟨cop 4596  dom cdm 5663  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is referenced by: (None)