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Theorem dffun7 6593
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 6594 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
Assertion
Ref Expression
dffun7 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun7
StepHypRef Expression
1 dffun6 6574 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 moabs 2543 . . . . . 6 (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
3 vex 3484 . . . . . . . 8 𝑥 ∈ V
43eldm 5911 . . . . . . 7 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
54imbi1i 349 . . . . . 6 ((𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦) ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
62, 5bitr4i 278 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
76albii 1819 . . . 4 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
8 df-ral 3062 . . . 4 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
97, 8bitr4i 278 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)
109anbi2i 623 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
111, 10bitri 275 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wex 1779  wcel 2108  ∃*wmo 2538  wral 3061   class class class wbr 5143  dom cdm 5685  Rel wrel 5690  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563
This theorem is referenced by:  dffun8  6594  dffun9  6595  brdom5  10569  imasaddfnlem  17573  imasvscafn  17582  funressnfv  47055
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