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Theorem funeu 4852
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funeu ((Fun 𝐹 A𝐹B) → ∃!y A𝐹y)
Distinct variable groups:   y,A   y,𝐹
Allowed substitution hint:   B(y)

Proof of Theorem funeu
StepHypRef Expression
1 funrel 4845 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 4496 . . . 4 ((Rel 𝐹 A𝐹B) → A dom 𝐹)
31, 2sylan 267 . . 3 ((Fun 𝐹 A𝐹B) → A dom 𝐹)
4 eldmg 4457 . . . 4 (A dom 𝐹 → (A dom 𝐹y A𝐹y))
54ibi 165 . . 3 (A dom 𝐹y A𝐹y)
63, 5syl 14 . 2 ((Fun 𝐹 A𝐹B) → y A𝐹y)
7 funmo 4843 . . . 4 (Fun 𝐹∃*y A𝐹y)
87adantr 261 . . 3 ((Fun 𝐹 A𝐹B) → ∃*y A𝐹y)
9 df-mo 1886 . . 3 (∃*y A𝐹y ↔ (y A𝐹y∃!y A𝐹y))
108, 9sylib 127 . 2 ((Fun 𝐹 A𝐹B) → (y A𝐹y∃!y A𝐹y))
116, 10mpd 13 1 ((Fun 𝐹 A𝐹B) → ∃!y A𝐹y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1362   wcel 1374  ∃!weu 1882  ∃*wmo 1883   class class class wbr 3738  dom cdm 4272  Rel wrel 4277  Fun wfun 4823
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-fun 4831
This theorem is referenced by:  funeu2  4853  funbrfv  5137
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