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Theorem fneu2 5277
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 5276 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
2 df-br 3968 . . 3 (𝐵𝐹𝑦 ↔ ⟨𝐵, 𝑦⟩ ∈ 𝐹)
32eubii 2015 . 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  ∃!weu 2006  wcel 2128  cop 3564   class class class wbr 3967   Fn wfn 5167
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-fun 5174  df-fn 5175
This theorem is referenced by:  feu  5354
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