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Theorem funmo 5138
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 5137 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
21simplbi 272 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex 4579 . . . . . 6 ((Rel 𝐹𝐴𝐹𝑦) → 𝐴 ∈ V)
43ex 114 . . . . 5 (Rel 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
52, 4syl 14 . . . 4 (Fun 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
65ancrd 324 . . 3 (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
76alrimiv 1846 . 2 (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8 breq1 3932 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98mobidv 2035 . . . . . 6 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
109imbi2d 229 . . . . 5 (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)))
111simprbi 273 . . . . . 6 (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
121119.21bi 1537 . . . . 5 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
1310, 12vtoclg 2746 . . . 4 (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))
1413com12 30 . . 3 (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
15 moanimv 2074 . . 3 (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
1614, 15sylibr 133 . 2 (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
17 moim 2063 . 2 (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
187, 16, 17sylc 62 1 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329   = wceq 1331  wcel 1480  ∃*wmo 2000  Vcvv 2686   class class class wbr 3929  Rel wrel 4544  Fun wfun 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125
This theorem is referenced by:  funeu  5148  funco  5163  imadif  5203  fneu  5227  dff3im  5565  shftfn  10608
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