Theorem 0fv 5665

Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Assertion

Ref	Expression
0fv	|- ( (/) ` A ) = (/)

Proof of Theorem 0fv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5326	. 2 |- ( (/) ` A ) = ( iota x A (/) x )
2		noel 3495	. . . . . 6 |- -. <. A , x >. e. (/)
3		df-br 4084	. . . . . 6 |- ( A (/) x <-> <. A , x >. e. (/) )
4	2, 3	mtbir 675	. . . . 5 |- -. A (/) x
5	4	nex 1546	. . . 4 |- -. E. x A (/) x
6		euex 2107	. . . 4 |- ( E! x A (/) x -> E. x A (/) x )
7	5, 6	mto 666	. . 3 |- -. E! x A (/) x
8		iotanul 5294	. . 3 |- ( -. E! x A (/) x -> ( iota x A (/) x ) = (/) )
9	7, 8	ax-mp 5	. 2 |- ( iota x A (/) x ) = (/)
10	1, 9	eqtri 2250	1 |- ( (/) ` A ) = (/)

Syntax hints:   -. wn 3   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   (/)c0 3491   <.cop 3669   class class class wbr 4083   iotacio 5276   ` cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326

This theorem is referenced by:  fv2prc  5666  ccat1st1st  11172  strsl0  13081