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Theorem funeu 4869
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 4862 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 4512 . . . 4 ((Rel 𝐹 A𝐹B) → A dom 𝐹)
31, 2sylan 267 . . 3 ((Fun 𝐹 A𝐹B) → A dom 𝐹)
4 eldmg 4473 . . . 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 4860 . . . 4 (Fun 𝐹∃*y A𝐹y)
87adantr 261 . . 3 ((Fun 𝐹 A𝐹B) → ∃*y A𝐹y)
9 df-mo 1901 . . 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 1378   wcel 1390  ∃!weu 1897  ∃*wmo 1898   class class class wbr 3755  dom cdm 4288  Rel wrel 4293  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:  funeu2  4870  funbrfv  5155
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