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Theorem funfveu 5131
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
funfveu ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)
Distinct variable groups:   y,A   y,𝐹

Proof of Theorem funfveu
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = A → (x dom 𝐹A dom 𝐹))
21anbi2d 437 . . . 4 (x = A → ((Fun 𝐹 x dom 𝐹) ↔ (Fun 𝐹 A dom 𝐹)))
3 breq1 3758 . . . . 5 (x = A → (x𝐹yA𝐹y))
43eubidv 1905 . . . 4 (x = A → (∃!y x𝐹y∃!y A𝐹y))
52, 4imbi12d 223 . . 3 (x = A → (((Fun 𝐹 x dom 𝐹) → ∃!y x𝐹y) ↔ ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)))
6 dffun8 4872 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 x dom 𝐹∃!y x𝐹y))
76simprbi 260 . . . 4 (Fun 𝐹x dom 𝐹∃!y x𝐹y)
87r19.21bi 2401 . . 3 ((Fun 𝐹 x dom 𝐹) → ∃!y x𝐹y)
95, 8vtoclg 2607 . 2 (A dom 𝐹 → ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y))
109anabsi7 515 1 ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  ∃!weu 1897  wral 2300   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-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-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847
This theorem is referenced by:  funfvex  5135
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