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Theorem dffun7 4955
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 4956 shows that it doesn't 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 4943 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 moabs 1965 . . . . . 6 (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
3 vex 2577 . . . . . . . 8 𝑥 ∈ V
43eldm 4559 . . . . . . 7 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
54imbi1i 231 . . . . . 6 ((𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦) ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦))
62, 5bitr4i 180 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
76albii 1375 . . . 4 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
8 df-ral 2328 . . . 4 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦))
97, 8bitr4i 180 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)
109anbi2i 438 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
111, 10bitri 177 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wex 1397  wcel 1409  ∃*wmo 1917  wral 2323   class class class wbr 3791  dom cdm 4372  Rel wrel 4377  Fun wfun 4923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-cnv 4380  df-co 4381  df-dm 4382  df-fun 4931
This theorem is referenced by:  dffun8  4956  dffun9  4957  funco  4967  funimaexglem  5009  frecuzrdgfn  9361
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