Theorem fnsnr 6921

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 6460	. . . 4 ⊢ (𝐹 Fn {𝐴} → (𝐹 ↾ {𝐴}) = 𝐹)
2		fnfun 6447	. . . . 5 ⊢ (𝐹 Fn {𝐴} → Fun 𝐹)
3		funressn 6915	. . . . 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 4005	. . 3 ⊢ (𝐹 Fn {𝐴} → 𝐹 ⊆ {⟨𝐴, (𝐹‘𝐴)⟩})
6	5	sseld 3965	. 2 ⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 ∈ {⟨𝐴, (𝐹‘𝐴)⟩}))
7		elsni 4577	. 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 1533   ∈ wcel 2110   ⊆ wss 3935  {csn 4560  ⟨cop 4566   ↾ cres 5551  Fun wfun 6343   Fn wfn 6344  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357

This theorem is referenced by:  fnsnb  6922  fnsnbt  39113