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

Theorem funmo 6583
Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.)
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 6576 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
21simplbi 497 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex1 5742 . . . . . 6 ((Rel 𝐹𝐴𝐹𝑦) → 𝐴 ∈ V)
43ex 412 . . . . 5 (Rel 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
52, 4syl 17 . . . 4 (Fun 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
65ancrd 551 . . 3 (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
76alrimiv 1925 . 2 (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
81simprbi 496 . . . 4 (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
9 breq1 5151 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
109mobidv 2547 . . . . 5 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
1110spcgv 3596 . . . 4 (𝐴 ∈ V → (∀𝑥∃*𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦))
128, 11syl5com 31 . . 3 (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
13 moanimv 2617 . . 3 (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
1412, 13sylibr 234 . 2 (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
15 moim 2542 . 2 (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
167, 14, 15sylc 65 1 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  Vcvv 3478   class class class wbr 5148  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  funeu  6593  funco  6608  fununmo  6615  imadif  6652  fneu  6679  dff3  7120  shftfn  15109
  Copyright terms: Public domain W3C validator