Theorem dffun7 5619

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 5620 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Assertion

Ref	Expression
dffun7

Distinct variable group:   ,,

Proof of Theorem dffun7

Step	Hyp	Ref	Expression
1		dffun6 5608	. 2
2		moabs 2315	. . . . . 6
3		vex 3112	. . . . . . . 8
4	3	eldm 5205	. . . . . . 7
5	4	imbi1i 325	. . . . . 6
6	2, 5	bitr4i 252	. . . . 5
7	6	albii 1640	. . . 4
8		df-ral 2812	. . . 4
9	7, 8	bitr4i 252	. . 3
10	9	anbi2i 694	. 2
11	1, 10	bitri 249	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  E*wmo 2283  A.wral 2807   class class class wbr 4452  domcdm 5004  Relwrel 5009  Funwfun 5587

This theorem is referenced by:  dffun8  5620  dffun9  5621  brdom5  8928  imasaddfnlem  14925  imasvscafn  14934  funressnfv  32213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-cnv 5012  df-co 5013  df-dm 5014  df-fun 5595