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Theorem fneu2 5198
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 5197 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
2 df-br 3900 . . 3 (𝐵𝐹𝑦 ↔ ⟨𝐵, 𝑦⟩ ∈ 𝐹)
32eubii 1986 . 2 (∃!𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
41, 3sylib 121 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  ∃!weu 1977  cop 3500   class class class wbr 3899   Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  feu  5275
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