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Theorem funeu2 4870
Description: There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
funeu2 ((Fun 𝐹 A, B 𝐹) → ∃!yA, y 𝐹)
Distinct variable groups:   y,A   y,𝐹
Allowed substitution hint:   B(y)

Proof of Theorem funeu2
StepHypRef Expression
1 df-br 3756 . 2 (A𝐹B ↔ ⟨A, B 𝐹)
2 funeu 4869 . . 3 ((Fun 𝐹 A𝐹B) → ∃!y A𝐹y)
3 df-br 3756 . . . 4 (A𝐹y ↔ ⟨A, y 𝐹)
43eubii 1906 . . 3 (∃!y A𝐹y∃!yA, y 𝐹)
52, 4sylib 127 . 2 ((Fun 𝐹 A𝐹B) → ∃!yA, y 𝐹)
61, 5sylan2br 272 1 ((Fun 𝐹 A, B 𝐹) → ∃!yA, y 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  ∃!weu 1897  cop 3370   class class class wbr 3755  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847
This theorem is referenced by:  funssres  4885
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