Theorem fnsnr 7019

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 6535	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2		fnfun 6517	. . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
3		funressn 7013	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
4	2, 3	syl 17	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
5	1, 4	eqsstrrd 3956	. . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
6	5	sseld 3916	. 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
7		elsni 4575	. 2 ⊢ (𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩} → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩)
8	6, 7	syl6 35	1 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  {csn 4558  ⟨cop 4564   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  fnsnb  7020  fnsnbt  40134