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Theorem fnsnb 6312
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 fnresdm 5897 . . . . . . . 8 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2 fnfun 5885 . . . . . . . . 9 (𝐹 Fn {𝐴} → Fun 𝐹)
3 funressn 6306 . . . . . . . . 9 (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
42, 3syl 17 . . . . . . . 8 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
51, 4eqsstr3d 3599 . . . . . . 7 (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹𝐴)⟩})
65sseld 3563 . . . . . 6 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
7 elsni 4138 . . . . . 6 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} → 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
86, 7syl6 34 . . . . 5 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
9 df-fn 5790 . . . . . . . 8 (𝐹 Fn {𝐴} ↔ (Fun 𝐹 ∧ dom 𝐹 = {𝐴}))
10 fnsnb.1 . . . . . . . . . . 11 𝐴 ∈ V
1110snid 4151 . . . . . . . . . 10 𝐴 ∈ {𝐴}
12 eleq2 2673 . . . . . . . . . 10 (dom 𝐹 = {𝐴} → (𝐴 ∈ dom 𝐹𝐴 ∈ {𝐴}))
1311, 12mpbiri 246 . . . . . . . . 9 (dom 𝐹 = {𝐴} → 𝐴 ∈ dom 𝐹)
1413anim2i 590 . . . . . . . 8 ((Fun 𝐹 ∧ dom 𝐹 = {𝐴}) → (Fun 𝐹𝐴 ∈ dom 𝐹))
159, 14sylbi 205 . . . . . . 7 (𝐹 Fn {𝐴} → (Fun 𝐹𝐴 ∈ dom 𝐹))
16 funfvop 6219 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
1715, 16syl 17 . . . . . 6 (𝐹 Fn {𝐴} → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
18 eleq1 2672 . . . . . 6 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1917, 18syl5ibrcom 235 . . . . 5 (𝐹 Fn {𝐴} → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
208, 19impbid 200 . . . 4 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
21 velsn 4137 . . . 4 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
2220, 21syl6bbr 276 . . 3 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
2322eqrdv 2604 . 2 (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
24 fvex 6095 . . . 4 (𝐹𝐴) ∈ V
2510, 24fnsn 5843 . . 3 {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}
26 fneq1 5876 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2725, 26mpbiri 246 . 2 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴})
2823, 27impbii 197 1 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  Vcvv 3169  wss 3536  {csn 4121  cop 4127  dom cdm 5025  cres 5027  Fun wfun 5781   Fn wfn 5782  cfv 5787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795
This theorem is referenced by:  fnprb  6352
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