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Theorem dffun5 5860
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 5858 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
2 df-br 4614 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32imbi1i 339 . . . . . 6 ((𝑥𝐴𝑦𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
43albii 1744 . . . . 5 (∀𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
54exbii 1771 . . . 4 (∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∃𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
65albii 1744 . . 3 (∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
76anbi2i 729 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
81, 7bitri 264 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wcel 1987  cop 4154   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:  funimaexg  5933  fvn0ssdmfun  6306  uzrdgfni  12697  dffrege115  37751
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