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Theorem fnsnb 7175
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, (𝐹𝐴)⟩})

Proof of Theorem fnsnb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnsnr 7174 . . . . 5 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
2 df-fn 6551 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3 fnsnb.1 . . . . . . . . . . 11 𝐴 ∈ V
43snid 4665 . . . . . . . . . 10 𝐴 ∈ {𝐴}
5 eleq2 2818 . . . . . . . . . 10 (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹𝐴 ∈ {𝐴}))
64, 5mpbiri 258 . . . . . . . . 9 (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
76anim2i 616 . . . . . . . 8 ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹𝐴 ∈ dom 𝐹))
82, 7sylbi 216 . . . . . . 7 (𝐹 Fn {𝐴} → (Fun 𝐹𝐴 ∈ dom 𝐹))
9 funfvop 7059 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
108, 9syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2817 . . . . . 6 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 246 . . . . 5 (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
131, 12impbid 211 . . . 4 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4645 . . . 4 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 289 . . 3 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2726 . 2 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
17 fvex 6910 . . . 4 (𝐹𝐴) ∈ V
183, 17fnsn 6611 . . 3 {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}
19 fneq1 6645 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2018, 19mpbiri 258 . 2 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴})
2116, 20impbii 208 1 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  {csn 4629  cop 4635  dom cdm 5678  Fun wfun 6542   Fn wfn 6543  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556
This theorem is referenced by:  fnprb  7220
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