Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun6 Structured version   Visualization version   GIF version

Theorem dffun6 5819
 Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
dffun6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem dffun6
StepHypRef Expression
1 nfcv 2751 . 2 𝑥𝐹
2 nfcv 2751 . 2 𝑦𝐹
31, 2dffun6f 5818 1 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃*wmo 2459   class class class wbr 4583  Rel wrel 5043  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  funmo  5820  dffun7  5830  fununfun  5848  funcnvsn  5850  funcnv2  5871  svrelfun  5875  fnres  5921  nfunsn  6135  dff3  6280  brdom3  9231  nqerf  9631  shftfn  13661  cnextfun  21678  perfdvf  23473  taylf  23919
 Copyright terms: Public domain W3C validator