Theorem dffun8 5916

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5915. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
dffun8	|- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) )

Distinct variable group:   x, y, A

Proof of Theorem dffun8

Step	Hyp	Ref	Expression
1		dffun7 5915	. 2 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
2		df-mo 2475	. . . . 5 |- ( E* y x A y <-> ( E. y x A y -> E! y x A y ) )
3		vex 3203	. . . . . . 7 |- x e. _V
4	3	eldm 5321	. . . . . 6 |- ( x e. dom A <-> E. y x A y )
5		pm5.5 351	. . . . . 6 |- ( E. y x A y -> ( ( E. y x A y -> E! y x A y ) <-> E! y x A y ) )
6	4, 5	sylbi 207	. . . . 5 |- ( x e. dom A -> ( ( E. y x A y -> E! y x A y ) <-> E! y x A y ) )
7	2, 6	syl5bb 272	. . . 4 |- ( x e. dom A -> ( E* y x A y <-> E! y x A y ) )
8	7	ralbiia 2979	. . 3 |- ( A. x e. dom A E* y x A y <-> A. x e. dom A E! y x A y )
9	8	anbi2i 730	. 2 |- ( ( Rel A /\ A. x e. dom A E* y x A y ) <-> ( Rel A /\ A. x e. dom A E! y x A y ) )
10	1, 9	bitri 264	1 |- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   A.wral 2912   class class class wbr 4653   dom cdm 5114   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-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-fun 5890

This theorem is referenced by:  dfdfat2  41211