Theorem fnsnr 7171

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 6661	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2		fnfun 6642	. . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
3		funressn 7166	. . . . 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 3944	. . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
6	5	sseld 3908	. 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
7		elsni 4597	. 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 1569   ∈ wcel 2144   ⊆ wss 3896  {csn 4584  ⟨cop 4834   ↾ cres 5652  Fun wfun 6535   Fn wfn 6536  ‘cfv 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549

This theorem is used by:  fnsnbg  7172