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Theorem fnsnbt 40660
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 7112 . . . . . . 7 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
21adantl 483 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
3 fnfun 6603 . . . . . . . 8 (𝐹 Fn {𝐴} → Fun 𝐹)
4 snidg 4621 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
54adantr 482 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6 fndm 6606 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
76adantl 483 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
85, 7eleqtrrd 2841 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9 funfvop 7001 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
103, 8, 9syl2an2 685 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2826 . . . . . . 7 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 247 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
132, 12impbid 211 . . . . 5 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4603 . . . . 5 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 289 . . . 4 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2735 . . 3 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1716ex 414 . 2 (𝐴 ∈ V → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
18 fvex 6856 . . . 4 (𝐹𝐴) ∈ V
19 fnsng 6554 . . . 4 ((𝐴 ∈ V ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
2018, 19mpan2 690 . . 3 (𝐴 ∈ V → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
21 fneq1 6594 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2220, 21syl5ibrcom 247 . 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 397   = wceq 1542  wcel 2107  Vcvv 3446  {csn 4587  cop 4593  dom cdm 5634  Fun wfun 6491   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  frlmsnic  40731
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