MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnsnr Structured version   Visualization version   GIF version

Theorem fnsnr 7156
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 6660 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2 fnfun 6640 . . . . 5 (𝐹 Fn {𝐴} → Fun 𝐹)
3 funressn 7150 . . . . 5 (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
42, 3syl 17 . . . 4 (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹𝐴)⟩})
51, 4eqsstrrd 4014 . . 3 (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹𝐴)⟩})
65sseld 3974 . 2 (𝐹 Fn {𝐴} → (𝐵𝐹𝐵 ∈ {⟨𝐴, (𝐹𝐴)⟩}))
7 elsni 4638 . 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 1533  wcel 2098  wss 3941  {csn 4621  cop 4627  cres 5669  Fun wfun 6528   Fn wfn 6529  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542
This theorem is referenced by:  fnsnb  7157  fnsnbt  41586
  Copyright terms: Public domain W3C validator