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Theorem fnsnr 6900
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 6439 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2 fnfun 6426 . . . . 5 (𝐹 Fn {𝐴} → Fun 𝐹)
3 funressn 6894 . . . . 5 (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
42, 3syl 17 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
51, 4eqsstrrd 3982 . . 3 (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹𝐴)⟩})
65sseld 3942 . 2 (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
7 elsni 4557 . 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 1538  wcel 2115  wss 3910  {csn 4540  cop 4546  cres 5530  Fun wfun 6322   Fn wfn 6323  cfv 6328
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  fnsnb  6901  fnsnbt  39246
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