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Theorem dffun7 4871
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 4872 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
Assertion
Ref Expression
dffun7  Fun  Rel  dom
Distinct variable group:   ,,

Proof of Theorem dffun7
StepHypRef Expression
1 dffun6 4859 . 2  Fun  Rel
2 moabs 1946 . . . . . 6
3 vex 2554 . . . . . . . 8 
_V
43eldm 4475 . . . . . . 7  dom
54imbi1i 227 . . . . . 6  dom
62, 5bitr4i 176 . . . . 5  dom
76albii 1356 . . . 4  dom
8 df-ral 2305 . . . 4  dom  dom
97, 8bitr4i 176 . . 3 
dom
109anbi2i 430 . 2  Rel  Rel 
dom
111, 10bitri 173 1  Fun  Rel  dom
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240  wex 1378   wcel 1390  wmo 1898  wral 2300   class class class wbr 3755   dom cdm 4288   Rel wrel 4293   Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847
This theorem is referenced by:  dffun8  4872  dffun9  4873  funco  4883  funimaexglem  4925  frecuzrdgfn  8879
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