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Theorem dffun5 6062
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 6060 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
2 df-br 4805 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32imbi1i 338 . . . . . 6 ((𝑥𝐴𝑦𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
43albii 1896 . . . . 5 (∀𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
54exbii 1923 . . . 4 (∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∃𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
65albii 1896 . . 3 (∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧) ↔ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧))
76anbi2i 732 . 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 383  wal 1630  wex 1853  wcel 2139  cop 4327   class class class wbr 4804  Rel wrel 5271  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-cnv 5274  df-co 5275  df-fun 6051
This theorem is referenced by:  funimaexg  6136  fvn0ssdmfun  6514  uzrdgfni  12971  dffrege115  38792
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