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Theorem fneu2 6665
Description: There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
Assertion
Ref Expression
fneu2 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu2
StepHypRef Expression
1 fneu 6664 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
2 df-br 5149 . . 3 (𝐵𝐹𝑦 ↔ ⟨𝐵, 𝑦⟩ ∈ 𝐹)
32eubii 2575 . 2 (∃!𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
41, 3sylib 217 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  ∃!weu 2558  cop 4635   class class class wbr 5148   Fn wfn 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6550  df-fn 6551
This theorem is referenced by:  feu  6773
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