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Theorem fnsnb 7186
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 7185 . . . . 5 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
2 df-fn 6564 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3 fnsnb.1 . . . . . . . . . . 11 𝐴 ∈ V
43snid 4662 . . . . . . . . . 10 𝐴 ∈ {𝐴}
5 eleq2 2830 . . . . . . . . . 10 (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹𝐴 ∈ {𝐴}))
64, 5mpbiri 258 . . . . . . . . 9 (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
76anim2i 617 . . . . . . . 8 ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹𝐴 ∈ dom 𝐹))
82, 7sylbi 217 . . . . . . 7 (𝐹 Fn {𝐴} → (Fun 𝐹𝐴 ∈ dom 𝐹))
9 funfvop 7070 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
108, 9syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2829 . . . . . 6 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 247 . . . . 5 (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
131, 12impbid 212 . . . 4 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4642 . . . 4 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 289 . . 3 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2735 . 2 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
17 fvex 6919 . . . 4 (𝐹𝐴) ∈ V
183, 17fnsn 6624 . . 3 {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}
19 fneq1 6659 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2018, 19mpbiri 258 . 2 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴})
2116, 20impbii 209 1 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  dom cdm 5685  Fun wfun 6555   Fn wfn 6556  cfv 6561
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fnprb  7228
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