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Theorem fnsnbt 40867
Description: A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)
Assertion
Ref Expression
fnsnbt (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Proof of Theorem fnsnbt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnsnr 7147 . . . . . . 7 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
21adantl 482 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
3 fnfun 6638 . . . . . . . 8 (𝐹 Fn {𝐴} → Fun 𝐹)
4 snidg 4656 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
54adantr 481 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6 fndm 6641 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
76adantl 482 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
85, 7eleqtrrd 2835 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9 funfvop 7036 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
103, 8, 9syl2an2 684 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2820 . . . . . . 7 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 246 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
132, 12impbid 211 . . . . 5 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4638 . . . . 5 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 288 . . . 4 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2729 . . 3 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1716ex 413 . 2 (𝐴 ∈ V → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
18 fvex 6891 . . . 4 (𝐹𝐴) ∈ V
19 fnsng 6589 . . . 4 ((𝐴 ∈ V ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
2018, 19mpan2 689 . . 3 (𝐴 ∈ V → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
21 fneq1 6629 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2220, 21syl5ibrcom 246 . 2 (𝐴 ∈ V → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴}))
2317, 22impbid 211 1 (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3473  {csn 4622  cop 4628  dom cdm 5669  Fun wfun 6526   Fn wfn 6527  cfv 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540
This theorem is referenced by:  frlmsnic  40914
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