MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funmo Structured version   Visualization version   GIF version

Theorem funmo 5873
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 5872 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
21simplbi 476 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex 5126 . . . . . 6 ((Rel 𝐹𝐴𝐹𝑦) → 𝐴 ∈ V)
43ex 450 . . . . 5 (Rel 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
52, 4syl 17 . . . 4 (Fun 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
65ancrd 576 . . 3 (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
76alrimiv 1852 . 2 (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8 breq1 4626 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98mobidv 2490 . . . . . 6 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
109imbi2d 330 . . . . 5 (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)))
111simprbi 480 . . . . . 6 (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
121119.21bi 2057 . . . . 5 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
1310, 12vtoclg 3256 . . . 4 (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))
1413com12 32 . . 3 (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
15 moanimv 2530 . . 3 (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
1614, 15sylibr 224 . 2 (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
17 moim 2518 . 2 (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
187, 16, 17sylc 65 1 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃*wmo 2470  Vcvv 3190   class class class wbr 4623  Rel wrel 5089  Fun wfun 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-fun 5859
This theorem is referenced by:  funeu  5882  funco  5896  fununmo  5901  imadif  5941  fneu  5963  dff3  6338  shftfn  13763
  Copyright terms: Public domain W3C validator