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Theorem dffun9 4873
 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 4871 . 2 (Fun A ↔ (Rel A x dom A∃*y xAy))
2 vex 2554 . . . . . . . 8 x V
3 vex 2554 . . . . . . . 8 y V
42, 3brelrn 4510 . . . . . . 7 (xAyy ran A)
54pm4.71ri 372 . . . . . 6 (xAy ↔ (y ran A xAy))
65mobii 1934 . . . . 5 (∃*y xAy∃*y(y ran A xAy))
7 df-rmo 2308 . . . . 5 (∃*y ran A xAy∃*y(y ran A xAy))
86, 7bitr4i 176 . . . 4 (∃*y xAy∃*y ran A xAy)
98ralbii 2324 . . 3 (x dom A∃*y xAyx dom A∃*y ran A xAy)
109anbi2i 430 . 2 ((Rel A x dom A∃*y xAy) ↔ (Rel A x dom A∃*y ran A xAy))
111, 10bitri 173 1 (Fun A ↔ (Rel A x dom A∃*y ran A xAy))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∃*wmo 1898  ∀wral 2300  ∃*wrmo 2303   class class class wbr 3755  dom cdm 4288  ran crn 4289  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-rmo 2308  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-rn 4299  df-fun 4847 This theorem is referenced by: (None)
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