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Theorem dffun9 6594
Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
dffun9 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun9
StepHypRef Expression
1 dffun7 6592 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2 vex 3483 . . . . . . . 8 𝑥 ∈ V
3 vex 3483 . . . . . . . 8 𝑦 ∈ V
42, 3brelrn 5952 . . . . . . 7 (𝑥𝐴𝑦𝑦 ∈ ran 𝐴)
54pm4.71ri 560 . . . . . 6 (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
65mobii 2547 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
7 df-rmo 3379 . . . . 5 (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
86, 7bitr4i 278 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
98ralbii 3092 . . 3 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
109anbi2i 623 . 2 ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
111, 10bitri 275 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2107  ∃*wmo 2537  wral 3060  ∃*wrmo 3378   class class class wbr 5142  dom cdm 5684  ran crn 5685  Rel wrel 5689  Fun wfun 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562
This theorem is referenced by:  brdom4  10571
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