Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun5 Structured version   Visualization version   GIF version

Theorem dffun5 5817
 Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun5 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun5
StepHypRef Expression
1 dffun3 5815 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
2 df-br 4584 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32imbi1i 338 . . . . . 6 ((𝑥𝐴𝑦𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
43albii 1737 . . . . 5 (∀𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
54exbii 1764 . . . 4 (∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∃𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
65albii 1737 . . 3 (∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
76anbi2i 726 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
81, 7bitri 263 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131   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:  funimaexg  5889  fvn0ssdmfun  6258  uzrdgfni  12619  dffrege115  37292
 Copyright terms: Public domain W3C validator