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Theorem funeu2 4849
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 3735 . 2 (A𝐹B ↔ ⟨A, B 𝐹)
2 funeu 4848 . . 3 ((Fun 𝐹 A𝐹B) → ∃!y A𝐹y)
3 df-br 3735 . . . 4 (A𝐹y ↔ ⟨A, y 𝐹)
43eubii 1887 . . 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 1370  ∃!weu 1878  cop 3349   class class class wbr 3734  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827
This theorem is referenced by:  funssres  4864
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