MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun3 Structured version   Visualization version   GIF version

Theorem dffun3 5858
Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun3 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun3
StepHypRef Expression
1 dffun2 5857 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
2 breq2 4617 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐴𝑦𝑥𝐴𝑧))
32mo4 2516 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
4 mo2v 2476 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
53, 4bitr3i 266 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
65albii 1744 . . 3 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
76anbi2i 729 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
81, 7bitri 264 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  ∃*wmo 2470   class class class wbr 4613  Rel wrel 5079  Fun wfun 5841
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 4741  ax-nul 4749  ax-pr 4867
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 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-cnv 5082  df-co 5083  df-fun 5849
This theorem is referenced by:  dffun5  5860  dffun6f  5861  sbcfung  5871  dffv2  6228
  Copyright terms: Public domain W3C validator