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Theorem funmo 4860
Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
Assertion
Ref Expression
funmo (Fun 𝐹∃*y A𝐹y)
Distinct variable groups:   y,A   y,𝐹

Proof of Theorem funmo
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dffun6 4859 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 x∃*y x𝐹y))
21simplbi 259 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex 4325 . . . . . 6 ((Rel 𝐹 A𝐹y) → A V)
43ex 108 . . . . 5 (Rel 𝐹 → (A𝐹yA V))
52, 4syl 14 . . . 4 (Fun 𝐹 → (A𝐹yA V))
65ancrd 309 . . 3 (Fun 𝐹 → (A𝐹y → (A V A𝐹y)))
76alrimiv 1751 . 2 (Fun 𝐹y(A𝐹y → (A V A𝐹y)))
8 breq1 3758 . . . . . . 7 (x = A → (x𝐹yA𝐹y))
98mobidv 1933 . . . . . 6 (x = A → (∃*y x𝐹y∃*y A𝐹y))
109imbi2d 219 . . . . 5 (x = A → ((Fun 𝐹∃*y x𝐹y) ↔ (Fun 𝐹∃*y A𝐹y)))
111simprbi 260 . . . . . 6 (Fun 𝐹x∃*y x𝐹y)
121119.21bi 1447 . . . . 5 (Fun 𝐹∃*y x𝐹y)
1310, 12vtoclg 2607 . . . 4 (A V → (Fun 𝐹∃*y A𝐹y))
1413com12 27 . . 3 (Fun 𝐹 → (A V → ∃*y A𝐹y))
15 moanimv 1972 . . 3 (∃*y(A V A𝐹y) ↔ (A V → ∃*y A𝐹y))
1614, 15sylibr 137 . 2 (Fun 𝐹∃*y(A V A𝐹y))
17 moim 1961 . 2 (y(A𝐹y → (A V A𝐹y)) → (∃*y(A V A𝐹y) → ∃*y A𝐹y))
187, 16, 17sylc 56 1 (Fun 𝐹∃*y A𝐹y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  ∃*wmo 1898  Vcvv 2551   class class class wbr 3755  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-fun 4847
This theorem is referenced by:  funeu  4869  funco  4883  imadif  4922  fneu  4946  dff3im  5255
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