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Theorem dffun9 5917
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 /\ A. x e. dom A E* y e. ran A x A y ) )
Distinct variable group:   x, y, A
Proof of Theorem dffun9
StepHypRef Expression
1 dffun7 5915 . 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2 vex 3203 . . . . . . . 8 |- x e. _V
3 vex 3203 . . . . . . . 8 |- y e. _V
42, 3brelrn 5356 . . . . . . 7 |- ( x A y -> y e. ran A )
54pm4.71ri 665 . . . . . 6 |- ( x A y <-> ( y e. ran A /\ x A y ) )
65mobii 2493 . . . . 5 |- ( E* y x A y <-> E* y ( y e. ran A /\ x A y ) )
7 df-rmo 2920 . . . . 5 |- ( E* y e. ran A x A y <-> E* y ( y e. ran A /\ x A y ) )
86, 7bitr4i 267 . . . 4 |- ( E* y x A y <-> E* y e. ran A x A y )
98ralbii 2980 . . 3 |- ( A. x e. dom A E* y x A y <-> A. x e. dom A E* y e. ran A x A y )
109anbi2i 730 . 2 |- ( ( Rel A /\ A. x e. dom A E* y x A y ) <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )
111, 10bitri 264 1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   E*wmo 2471   A.wral 2912   E*wrmo 2915   class class class wbr 4653   dom cdm 5114   ran crn 5115   Rel wrel 5119   Fun wfun 5882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890
This theorem is referenced by:  brdom4  9352
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