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Theorem fnsnr 7185
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 6687 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2 fnfun 6668 . . . . 5 (𝐹 Fn {𝐴} → Fun 𝐹)
3 funressn 7179 . . . . 5 (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
42, 3syl 17 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
51, 4eqsstrrd 4019 . . 3 (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹𝐴)⟩})
65sseld 3982 . 2 (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
7 elsni 4643 . 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 1540  wcel 2108  wss 3951  {csn 4626  cop 4632  cres 5687  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fnsnb  7186  fnsnbt  42271
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