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Theorem fnsnr 7112
Description: If a class belongs to a function on a singleton, then that class is the obvious ordered pair. Note that this theorem also holds when 𝐴 is a proper class, but its meaning is then different. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
Assertion
Ref Expression
fnsnr (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 = ⟨𝐴, (𝐹𝐴)⟩))

Proof of Theorem fnsnr
StepHypRef Expression
1 fnresdm 6621 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2 fnfun 6603 . . . . 5 (𝐹 Fn {𝐴} → Fun 𝐹)
3 funressn 7106 . . . . 5 (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
42, 3syl 17 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
51, 4eqsstrrd 3984 . . 3 (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹𝐴)⟩})
65sseld 3944 . 2 (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
7 elsni 4604 . 2 (𝐵 ∈ {⟨𝐴, (𝐹𝐴)⟩} → 𝐵 = ⟨𝐴, (𝐹𝐴)⟩)
86, 7syl6 35 1 (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 = ⟨𝐴, (𝐹𝐴)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wss 3911  {csn 4587  cop 4593  cres 5636  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:  fnsnb  7113  fnsnbt  40660
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