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Theorem dffun8 4872
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4871. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dffun8 (Fun A ↔ (Rel A x dom A∃!y xAy))
Distinct variable group:   x,y,A

Proof of Theorem dffun8
StepHypRef Expression
1 dffun7 4871 . 2 (Fun A ↔ (Rel A x dom A∃*y xAy))
2 df-mo 1901 . . . . 5 (∃*y xAy ↔ (y xAy∃!y xAy))
3 vex 2554 . . . . . . 7 x V
43eldm 4475 . . . . . 6 (x dom Ay xAy)
5 pm5.5 231 . . . . . 6 (y xAy → ((y xAy∃!y xAy) ↔ ∃!y xAy))
64, 5sylbi 114 . . . . 5 (x dom A → ((y xAy∃!y xAy) ↔ ∃!y xAy))
72, 6syl5bb 181 . . . 4 (x dom A → (∃*y xAy∃!y xAy))
87ralbiia 2332 . . 3 (x dom A∃*y xAyx dom A∃!y xAy)
98anbi2i 430 . 2 ((Rel A x dom A∃*y xAy) ↔ (Rel A x dom A∃!y xAy))
101, 9bitri 173 1 (Fun A ↔ (Rel A x dom A∃!y xAy))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  ∃!weu 1897  ∃*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-bnd 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:  funco  4883  funimaexglem  4925  funfveu  5131
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