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Theorem fneu2 4919
Description: There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
Assertion
Ref Expression
fneu2 ((𝐹 Fn A B A) → ∃!yB, y 𝐹)
Distinct variable groups:   y,𝐹   y,B
Allowed substitution hint:   A(y)

Proof of Theorem fneu2
StepHypRef Expression
1 fneu 4918 . 2 ((𝐹 Fn A B A) → ∃!y B𝐹y)
2 df-br 3729 . . 3 (B𝐹y ↔ ⟨B, y 𝐹)
32eubii 1883 . 2 (∃!y B𝐹y∃!yB, y 𝐹)
41, 3sylib 127 1 ((𝐹 Fn A B A) → ∃!yB, y 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1367  ∃!weu 1874  cop 3343   class class class wbr 3728   Fn wfn 4813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-fun 4820  df-fn 4821
This theorem is referenced by:  feu  4986
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