Theorem fnsnr 7165

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} → (𝐵 ∈ 𝐹 → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩))

Proof of Theorem fnsnr

Step	Hyp	Ref	Expression
1		fnresdm 6658	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2		fnfun 6639	. . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
3		funressn 7160	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
4	2, 3	syl 18	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
5	1, 4	eqsstrrd 3973	. . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
6	5	sseld 3937	. 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
7		elsni 4608	. 2 ⊢ (𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩)
8	6, 7	syl6 36	1 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  {csn 4591  ⟨cop 4597   ↾ cres 5665  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by:  fnsnbg  7166  fnsnbOLD  7168