Theorem fneu2 6597

Description: There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)

Assertion

Ref	Expression
fneu2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐵

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu2

Step	Hyp	Ref	Expression
1		fneu 6596	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)
2		df-br 5096	. . 3 ⊢ (𝐵𝐹𝑦 ↔ ⟨𝐵, 𝑦⟩ ∈ 𝐹)
3	2	eubii 2578	. 2 ⊢ (∃!𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)
4	1, 3	sylib 218	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∃!weu 2561  ⟨cop 4585   class class class wbr 5095   Fn wfn 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-fun 6488  df-fn 6489

This theorem is referenced by:  feu  6704