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Theorem dffun7 5212
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 5213 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 5199 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 moabs 2062 . . . . . 6 (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
3 vex 2727 . . . . . . . 8 𝑥 ∈ V
43eldm 4798 . . . . . . 7 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
54imbi1i 237 . . . . . 6 ((𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦) ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
62, 5bitr4i 186 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
76albii 1457 . . . 4 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
8 df-ral 2447 . . . 4 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
97, 8bitr4i 186 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)
109anbi2i 453 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
111, 10bitri 183 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1340  wex 1479  ∃*wmo 2014  wcel 2135  wral 2442   class class class wbr 3979  dom cdm 4601  Rel wrel 4606  Fun wfun 5179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-id 4268  df-cnv 4609  df-co 4610  df-dm 4611  df-fun 5187
This theorem is referenced by:  dffun8  5213  dffun9  5214  funco  5225  funimaexglem  5268  frecuzrdgtcl  10341  frecuzrdgfunlem  10348
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