Theorem fneu2 5444

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 5443	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)
2		df-br 4094	. . 3 ⊢ (𝐵𝐹𝑦 ↔ ⟨𝐵, 𝑦⟩ ∈ 𝐹)
3	2	eubii 2088	. 2 ⊢ (∃!𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)
4	1, 3	sylib 122	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 104  ∃!weu 2079   ∈ wcel 2202  ⟨cop 3676   class class class wbr 4093   Fn wfn 5328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-fun 5335  df-fn 5336

This theorem is referenced by:  feu  5527