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Theorem funmo 5649
 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 5648 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
21simplbi 469 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex 4920 . . . . . 6 ((Rel 𝐹𝐴𝐹𝑦) → 𝐴 ∈ V)
43ex 443 . . . . 5 (Rel 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
52, 4syl 17 . . . 4 (Fun 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
65ancrd 569 . . 3 (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
76alrimiv 1804 . 2 (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8 breq1 4437 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98mobidv 2374 . . . . . 6 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
109imbi2d 325 . . . . 5 (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)))
111simprbi 473 . . . . . 6 (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
121119.21bi 2000 . . . . 5 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
1310, 12vtoclg 3128 . . . 4 (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))
1413com12 32 . . 3 (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
15 moanimv 2414 . . 3 (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
1614, 15sylibr 219 . 2 (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
17 moim 2402 . 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 setvar class Syntax hints:   → wi 4   ∧ wa 378  ∀wal 1466   = wceq 1468   ∈ wcel 1937  ∃*wmo 2354  Vcvv 3066   class class class wbr 4434  Rel wrel 4885  Fun wfun 5627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pr 4680 This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-br 4435  df-opab 4494  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-fun 5635 This theorem is referenced by:  funeu  5657  funco  5671  fununmo  5676  imadif  5713  fneu  5735  dff3  6102  shftfn  13296
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