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Theorem dffun3 6563
Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.)
Assertion
Ref Expression
dffun3 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun3
StepHypRef Expression
1 dffun6 6562 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 df-mo 2528 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
32albii 1813 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
43anbi2i 621 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
51, 4bitri 274 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wex 1773  ∃*wmo 2526   class class class wbr 5149  Rel wrel 5683  Fun wfun 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-fun 6551
This theorem is referenced by:  dffun5  6566  dffun6f  6567  sbcfung  6578  dffv2  6992
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