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Theorem fnsnb 7157
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 7156 . . . . 5 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
2 df-fn 6537 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
3 fnsnb.1 . . . . . . . . . . 11 𝐴 ∈ V
43snid 4657 . . . . . . . . . 10 𝐴 ∈ {𝐴}
5 eleq2 2814 . . . . . . . . . 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 7042 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
108, 9syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2813 . . . . . 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 4637 . . . 4 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 289 . . 3 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2722 . 2 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
17 fvex 6895 . . . 4 (𝐹𝐴) ∈ V
183, 17fnsn 6597 . . 3 {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}
19 fneq1 6631 . . 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 1533  wcel 2098  Vcvv 3466  {csn 4621  cop 4627  dom cdm 5667  Fun wfun 6528   Fn wfn 6529  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542
This theorem is referenced by:  fnprb  7202
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