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Theorem fnsnbt 39969
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 7001 . . . . . . 7 (𝐹 Fn {𝐴} → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
21adantl 485 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
3 fnfun 6499 . . . . . . . 8 (𝐹 Fn {𝐴} → Fun 𝐹)
4 snidg 4591 . . . . . . . . . 10 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
54adantr 484 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ {𝐴})
6 fndm 6502 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
76adantl 485 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → dom 𝐹 = {𝐴})
85, 7eleqtrrd 2843 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐴 ∈ dom 𝐹)
9 funfvop 6891 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
103, 8, 9syl2an2 686 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
11 eleq1 2827 . . . . . . 7 (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → (𝑥𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
1210, 11syl5ibrcom 250 . . . . . 6 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥 = ⟨𝐴, (𝐹𝐴)⟩ → 𝑥𝐹))
132, 12impbid 215 . . . . 5 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 = ⟨𝐴, (𝐹𝐴)⟩))
14 velsn 4573 . . . . 5 (𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝑥 = ⟨𝐴, (𝐹𝐴)⟩)
1513, 14bitr4di 292 . . . 4 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → (𝑥𝐹𝑥 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
1615eqrdv 2737 . . 3 ((𝐴 ∈ V ∧ 𝐹 Fn {𝐴}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
1716ex 416 . 2 (𝐴 ∈ V → (𝐹 Fn {𝐴} → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
18 fvex 6751 . . . 4 (𝐹𝐴) ∈ V
19 fnsng 6452 . . . 4 ((𝐴 ∈ V ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
2018, 19mpan2 691 . . 3 (𝐴 ∈ V → {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴})
21 fneq1 6490 . . 3 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → (𝐹 Fn {𝐴} ↔ {⟨𝐴, (𝐹𝐴)⟩} Fn {𝐴}))
2220, 21syl5ibrcom 250 . 2 (𝐴 ∈ V → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} → 𝐹 Fn {𝐴}))
2317, 22impbid 215 1 (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  Vcvv 3423  {csn 4557  cop 4563  dom cdm 5568  Fun wfun 6394   Fn wfn 6395  cfv 6400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408
This theorem is referenced by:  frlmsnic  40023
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